Lecture 2 recap Th(A) computing Th(AU) and Th(Ainfin)from Th(A)

F(C ) the R-algebra of all formulas over a Clowastndashalgebra C Fn(C ) All ϕ isin F(C ) whose free variables included in x0 xnminus1If C le A ϕ(x) isin Fn(C ) and a isin An then ϕA(a) is theinterpretation of ϕ in A at a983042ϕ983042 = supAa |ϕA(a)| norms F(C)The theory of A Th(A) is the character on F0(C) defined byϕ 983041rarr ϕAA is an elementary submodel of B Q ≼ B if A le B andϕA(a) = ϕB(a) for all n all ϕ isin Fn(C) (equivalently for allϕ isin Fn(A)983040Losrsquos Theorem A ≼ AU Ghasemi Th(Ainfin) can be computed from Th(A)

Thanks to David Jekel for asking where exactly the proof of 983040Lostheorem breaks down for Ainfin

Example

For any nontrivial A define a isin Ainfin by an = 0 if n even and an = 1if n odd Let ϕ(x) = min(983042x983042 9830421minus x983042)Then ϕA(an) = 0 for all n but ϕAinfin(a) = 1

Massive Clowastndashalgebras Lecture 3Countable saturation and transfinite intertwining

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Thanks to David Jekel for asking where exactly the proof of 983040Lostheorem breaks down for Ainfin

Example

For any nontrivial A define a isin Ainfin by an = 0 if n even and an = 1if n odd Let ϕ(x) = min(983042x983042 9830421minus x983042)Then ϕA(an) = 0 for all n but ϕAinfin(a) = 1

Massive Clowastndashalgebras Lecture 3Countable saturation and transfinite intertwining

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Massive Clowastndashalgebras Lecture 3Countable saturation and transfinite intertwining

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Types

Suppose k ge 1 ϕ(x) is a formula in Fk(C ) and r isin R Anexpression of the form

983042ϕ(x0 xkminus1)983042 = r (1)

is a k-condition on x A set of k-conditions1 is a k-type over C We usually suppress the mention of k and say condition and type

Example (Concrete types)

Fix n A and a isin An Then the type of a in A typeA(a) is theevaluation

Fn(A) ni ϕ 983041rarr ϕA(a)

It is a character If n = 0 we get Th(A) computed in F0(A)

1Ie alsquosystem of equationsrsquo

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Abstract types

Think of Th(A) as a 0-type Also we can think of the theory of Aas a character (ϕ 983041rarr ϕA) or as its kernel This applies to typesA type over C is consistent if for every m isin N there existc(0) c(k minus 1) in C1 such that

maxjltm

|ϕCj (c(0) c(k minus 1))minus rj | lt 1m

It is realized in C if there exist c(0) c(k minus 1) in C1 such that

ϕCj (c(0) c(k minus 1)) = rj

for all j

Remark

1 That is C1 ndash the unit ball of C ndash not C

2 The formulas are evaluated in C

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Countable saturation

DefinitionA Clowastndashalgebra C is countably saturated if every countableconsistent type over C is realized in C1

LemmaA separable Clowastndashalgebra is countably saturated iff it isfinite-dimensional

ProoflArr Since the unit ball is norm-compact the partial realizationshave a convergent sequence The limit realizes the typeFor rArr let an n isin N be a countable dense subset of A The typewith conditions 983042x minus an983042 ge 1 is consistent but not realizedin A

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le m

By 983040Loslim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition

AU is countably saturated for every nonprincipal U on N andevery A

ProofFix a consistent type ϕj(x) = rj for j isin NFix c(m) for m isin N such that |ϕj(c(m))minus rj | lt 1m for all j le mBy 983040Los

lim supnrarrU

|ϕAj (c(m)n)minus rj | le 1m

Find N = X0 sup X1 sup X2 sup in U such that|ϕA

j (c(m)n)minus rj | le 1m for all n isin Xm We may assume983127n Xn = empty

Define c in AU by cn = c(m)n if n isin Xm Xm+1 Fix j By 983040Los for every j

ϕAUj (c) = lim

nrarrUϕAj (cn) = rj

Thus c realizes the type

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Kirchbergrsquos ε-test

Singlehandedly and independently from model-theorists EKirchberg reinvented countable saturation

Kirchberg ε-test

If f mn Xn rarr [0infin) for m n in N and for every m there existsx(m) isin

983124N Xn such that

maxnlem

f mn (x(m)) le 1n

then there exists x isin983124

n Xn such that

supn

limmrarrU

f mn (x) = 0

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall Φ isin Aut(A) implies ΦU isin Aut(AU )Some Φ isin Aut(A) is approximately inner if it is a point-norm limitof a net of inner automorphisms

Proposition

Suppose Φ isin Aut(A) and Φ is approximately innerThen the restriction of ΦU to any separable subalgebra B of AU isimplemented by a unitary in AU

ProofConsider the type t consisting of 983042xxlowast minus 1983042 = 0 983042xlowastx minus 1983042 = 0983042xbxlowast minus ΦU (b)983042 = 02 for a dense set of b isin B Then t is consistent because Φ is approximately innerIf c realizes t then it is a unitary that implements ΦU on B

Note that ΦU is inner iff Φ is inner (by 983040Los applied to (AΦ))

2For the purists I should have introduced a function symbol to denote Φand yadda yadda yadda makes no difference

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B

Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Recall if ϕ is a (pure) state of A then ϕU is a (pure) state of AU

Proposition

If ϕ is a pure state on A and B is a separable Clowastndashsubalgebra ofAU then there exists a isin A+ 983042a983042 = 1 ϕU (a) = 1 and

aba = ϕ(b)a2

for all b isin B

ProofConsider the type consisting of the conditions 983042xlowastx983042 = 1ϕU (x

lowastx) = 1 983042xlowastxbxlowastx minus ϕ(b)xlowastx983042 = 0 for a dense set of b isin B Since ϕ can be excised3 this type is consistent If c realizes it leta = clowastc

If A has real rank zero we can choose a to be a projection

3If you donrsquot know what this means It is exactly the assertion that thistype is consistent and a theorem of AkemannndashAndersonndashPedersen

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

More of the same

Suppose C is AU or Ainfin

1 C is SAWlowast If B0 and B1 are separable Clowastndashsubalgebras of C and b0b1 = 0 for all bj isin Bj then there exists c isin (C )+ suchthat b0c = b0 and cb1 = 0 for all bj isin Bj

2 (FndashChoindashOzawa) Every uniformly bounded representation ofan amenable group into C is unitarizable

3 If a isin C is normal U sube Sp(a) is open and g U rarr C iscontinuous (not necessarily bounded) then there exists c isin Csuch that fg(a) = f (c) for every f isin C0(U) If f isreal-valued we can choose c to be self-adjoint (The BDFlsquoSecond Splitting Lemmarsquo is a special case)

All results on this slide hold in the corona of every σ-unitalClowastndashalgebra (FndashHart lsquocountable degree-1 saturationrsquo)

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower bound

Consider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

A reduced product which is not countably saturated

Example

Let Zdensity zero = X sube N| lim supn |X cap n|n = 0 Then for anyunital A there is a countable consistent type over AZdensity zero

thatis not realized in AZdensity zero

Let X (k) = m isin N m mod 2k = 0 Thenlim |X (k) cap n|n = 2minusk and pX (k) for k isin N is a decreasingsequence of nonzero projections in AZdensityzero

with no nonzerolower boundConsider the type 983042y minus ylowast983042 = 0 983042y2 minus y983042 = 0 983042y983042 = 1983042pX (k)y minus y983042 = 0 for all k Every finite subset is realized by somepX (k) but the type is not realized in AZdensity zero

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Theorem (FndashShelah)

Every asymptotic sequence algebra eg AFinlowast and983124

n An983119

n Anis countably saturated

The proof uses Ghasemirsquos Theorem instead of 983040Losrsquos Theorem

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

The diagonal embedding and the relative commutant

DefinitionThe diagonal embedding ι A rarr AF is defined by

ι(a) = (a a a )

We identify A with ι[A] and think of AF as an extension of AThe relative commutant of A in AF is

AF cap Aprime = b isin AF [a b] = 0 for all a isin A

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar)

For every n ge 1 letψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr BU cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr BU cap B prime

Proof We first define a type tA(x) in F0(C) with alefsym0 variables thatdescribes a dense subset (an) of A up to isomorphismFix a uniformly dense sequence of lowast-polynomials over C innon-commuting variables Qn(x0 xnminus1) Let ξk(x) be theformula

|983042Qk(x0 xnminus1)983042 minus 983042QAk (a0 anminus1)983042|

(note that QAk ( ) is a scalar) For every n ge 1 let

ψAn(y0 ynminus1) denote

infxmaxi jltn

983042[yi xj ]983042+maxkltn

ξk(x)

LemmaA rarr BU cap B prime iff supy ψ

BAn(y) = 0

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proof that A rarr BU cap B prime implies983121alefsym1

max A rarr BU cap B prime contrsquod

Fix a separable X sube BU cap B prime and the type that lsquosaysrsquo that there isan isomorphic copy of A in BU cap (B cap X )primeThis type is consistent for any countable subset X of BU Now go(alefsym1 times)The same proof gives the following

Proposition (Kirchberg)

Suppose AB are separable and unital and Φ A rarr Binfin cap B prime is aunital lowast-homomorphism Then there is a unital lowast-homomorphismΨ

983121alefsym1max A rarr Binfin cap B prime

(It works whenever BF is countably saturated)

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Tensorial absorptionA unital Clowastndashalgebra D has an approximately inner half-flip if themaps from D to D otimesmax D

Φl(d) = d otimes 1 Φr (d) = 1otimes d

are approximately unitarily equivalent

Examples

Mn(C) any UHF algebra O2 Oinfin any Kirchberg algebra inCuntz standard form Z all have approximately inner half-flip

Theorem (McDuff EffrosndashRosenberg TomsndashWinter)

If A is separable and unital and D has an approximately inner halfflip the following are equivalent

1 Aotimes D sim= A

2 D unitally embeds into Ainfin cap Aprime

3 D unitally embeds into AU cap Aprime for some (any) U

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Lemma (Elliott Roslashrdam)

If A sube B are separable and unital (same unit) and someun isin U(BU cap Aprime) for n isin N satisfy limn dist((Ad un)(b)AU ) = 0then A sim= B

ProofFix an for n isin N dense in A and bn for n isin N dense in B Bylifting um and diagonalizing find un isin U(B) and anj so that withwn = u1u2 un for all j le n we have

983042[un aj ]983042 lt 2minusn 983042wlowastnbjwn minus anj983042 lt 1n 983042[un akj ]983042 lt 2minusn

Let Ψn(a) = (Adwn)(a) for a isin A and n isin N For j lt n we have

Ψn+1(aj) = Ψn(unajulowastn) asymp2minusn Ψn(aj)

and the pointwise limit of the Ψnrsquos Ψ A rarr B is an injectivelowast-homomorphism Since 983042wlowast

nbjwn minus anj983042 lt 1n we havelimn(Ψn(anj)) = bj hence Ψ is surjective

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Proof that D rarr AU cap Aprime implies Aotimes D sim= A

FirstD rarr AU cap Aprime rarr (Aotimes D)U cap Aprime

Since D has the approximately inner half-flip by countablesaturation there is a unitary u isin (Aotimes D)U cap Aprime that flips the twocopies of D Since D rarr AU cap Aprime we can lift u to (un) in AU cap Aprime

such that limn(undulowastnA) rarr 0 now apply the intertwining lemma

D rarr Ainfin cap Aprime implies Aotimes D sim= A the same proof

To prove the converse(s) Countable saturation

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

By R we denote the hyperfinite II1 factor with separable predual

Theorem (McDuff)

For a II1 factor M with a separable predual M2(C) rarr MU capM prime ifand only if MotimesR sim= M

This predates the EffrosndashRosenberg theorem for which it served asan inspiration

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Tracial ultrapowers

Suppose A has a tracial state (and it is unital) The uniform tracialnorm on A is

983042a9830422u = supτisinT (A)

τ(alowasta)12

It extends to a seminorm 983042 middot 983042U2u on AU The trace kernel ideal is

J = a isin AU |983042a983042U2u = 0

Let AU = AUJ

RemarkAU is the ultrapower of the structure (A 983042 middot 9830422u) This is A withthe uniform tracial norm (The sorts are still operatornorm-bounded balls)It is therefore countably saturated but note that the operatornorm is not a part of this structure

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Example

If A is simple nuclear and monotracial then AU = AUJ sim= RU where R is the hyperfinite factorOzawarsquos lsquoWlowastndashbundlesrsquo and lsquostrict closuresrsquo

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Theorem (Sato KirchbergndashRoslashrdam)

Suppose T (A) ∕= empty A unital B le AU separable Then thequotient map πJ maps AU cap B prime onto AU cap πJ [B]

prime

ProofClearly πJ [AU cap B prime] sube AU cap πJ [B]

primeFix c isin AU such that πJ(c) isin πJ [B]

primeConsider the type consisting of conditions983042x983042 = 1 983042xlowastx9830422u = 0 983042[x c]983042 = 0 983042[x b]983042 = 0983042[b c](1minus xlowastx)983042 = 0 for a dense set of b isin B Since J has a quasicentral approximate unit (Arveson) this type isconsistentBy countable saturation of AU some d realizes this type Thendlowastd isin J+ and dlowastd isin AU cap bn c |n isin Nprime Thereforec1 = (1minus dlowastd)c satisfies πJ(c1) = πJ(c) and[bn c1] = [bn c](1minus dlowastd) hence πJ(c1) isin πJ [B]

prime

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin

Next time (tomorrow)

1 lsquo983040Los + countable saturationrsquo is all you need to know about AU

2 What about Ainfin