Lecture 1 recap

Given A and a filter F letcF (An) = (an) isin ℓinfin| lim supnrarrF 983042an983042 = 0The reduced product of A is

AF = ℓinfin(A)cF (A)

AFinlowast denoted Ainfin is the asymptotic sequence algebraIf F is an ultrafilter then AF is an ultrapower of AπF ℓinfin(A) rarr AF is the quotient mapWe routinely identify an element of AF with its representingsequence (an)If ϕ is a state of A then ϕU is a state of AU Ditto forautomorphisms Think of (AU ϕU ) as (Aϕ)U

More Lecture 1 recap

Question (Updated thanks to Sam Evington)

Assume the Continuum Hypothesis Find simple separablenonisomorphic AF algebras A and B such that AU and BU areisomorphicAre there such A and B even if the Continuum Hypothesis fails

The same questions for Kirchberg algebras

Massive Clowastndashalgebras Lecture 2What is preserved by taking an ultrapower

and why

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

We fix a nonprincipal ultrafilter U on N throughout

Proposition (GendashHadwin)

There exists a simple Clowastndashalgebra A such that AU is not simple

ProofLet A be a UHF algebra We only need a nontrivial tracial state τon A such that for every ε gt 0 there exists a isin A+ 983042a983042 = 1 andτ(a) lt εThen τU isin T (AU ) and a isin AU τ(alowasta) = 0 is a nontrivialideal

Proposition (Kirchberg)

For any A AU is exact if and only if A is subhomogeneous

For rArr we use the following

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)Once proven this will imply that A has the (non-exact) Clowastndashalgebra983124

U Mn(C) as a subquotient

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

More Lecture 1 recap

Question (Updated thanks to Sam Evington)

Assume the Continuum Hypothesis Find simple separablenonisomorphic AF algebras A and B such that AU and BU areisomorphicAre there such A and B even if the Continuum Hypothesis fails

The same questions for Kirchberg algebras

Massive Clowastndashalgebras Lecture 2What is preserved by taking an ultrapower

and why

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

We fix a nonprincipal ultrafilter U on N throughout

Proposition (GendashHadwin)

There exists a simple Clowastndashalgebra A such that AU is not simple

ProofLet A be a UHF algebra We only need a nontrivial tracial state τon A such that for every ε gt 0 there exists a isin A+ 983042a983042 = 1 andτ(a) lt εThen τU isin T (AU ) and a isin AU τ(alowasta) = 0 is a nontrivialideal

Proposition (Kirchberg)

For any A AU is exact if and only if A is subhomogeneous

For rArr we use the following

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)Once proven this will imply that A has the (non-exact) Clowastndashalgebra983124

U Mn(C) as a subquotient

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Massive Clowastndashalgebras Lecture 2What is preserved by taking an ultrapower

and why

Ilijas Farah

YMClowastAYWlowastCA Copenhagen August 2019

We fix a nonprincipal ultrafilter U on N throughout

Proposition (GendashHadwin)

There exists a simple Clowastndashalgebra A such that AU is not simple

ProofLet A be a UHF algebra We only need a nontrivial tracial state τon A such that for every ε gt 0 there exists a isin A+ 983042a983042 = 1 andτ(a) lt εThen τU isin T (AU ) and a isin AU τ(alowasta) = 0 is a nontrivialideal

Proposition (Kirchberg)

For any A AU is exact if and only if A is subhomogeneous

For rArr we use the following

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)Once proven this will imply that A has the (non-exact) Clowastndashalgebra983124

U Mn(C) as a subquotient

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

We fix a nonprincipal ultrafilter U on N throughout

Proposition (GendashHadwin)

There exists a simple Clowastndashalgebra A such that AU is not simple

ProofLet A be a UHF algebra We only need a nontrivial tracial state τon A such that for every ε gt 0 there exists a isin A+ 983042a983042 = 1 andτ(a) lt εThen τU isin T (AU ) and a isin AU τ(alowasta) = 0 is a nontrivialideal

Proposition (Kirchberg)

For any A AU is exact if and only if A is subhomogeneous

For rArr we use the following

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)Once proven this will imply that A has the (non-exact) Clowastndashalgebra983124

U Mn(C) as a subquotient

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Proposition (Kirchberg)

For any A AU is exact if and only if A is subhomogeneous

For rArr we use the following

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)Once proven this will imply that A has the (non-exact) Clowastndashalgebra983124

U Mn(C) as a subquotient

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

LemmaIf π A rarr B(H) is irreducible and dim(H) ge n then there areB le A and J ⊳ B such that BJ sim= Mn(C)

ProofFix K le H such that dim(K ) = n so that B(K ) sim= Mn(C) Let

J = a isin A pKa = apk = 0

By the Kadison Transitivity Theorem for u isin U(K ) there existsu isin U(A) such that pK upK = uThen upK = pK u = uIf F ⋐ U(K ) generates B(K ) then B = Clowast(u u isin F cup J) is asrequired

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

What properties of A are preserved in AU

Some of the following are exercises

property A rarr AU A rarr Ainfin

separable No Nononseparable Yes Yes

simple No Nonon-simple Yes Yesnuclear No Noexact No No

UHF AF AT AI A-whatever No Noprimitive Yes Nounital Yes Yes

projectionless Yes Yes

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

But what properties of A are preserved in AU

Q(x) A lowast-polynomial with coefficients in AU in non-commutingvariables(an) a representing sequence of a isin AU

983042Q(a)983042 = limnrarrU

983042Q(an)983042

If f R rarr R is continuous then

f (983042Q(a)983042) = limnrarrU

f (983042Q(an)983042)

supxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

supxisinA983042x983042le1

f (983042Q(an x)983042)

infxisinAU 983042x983042le1

f (983042Q(a x)983042) = limnrarrU

infxisinA983042x983042le1

f (983042Q(an x)983042)

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

The previous slide gives us a hint on what properties of theelements of A and A itself are preserved by ultrapowers

We only need to formalize what just happened

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Syntax Language Formulas Sorry this is necessary

Let F(C ) denote the set of all formulas over a Clowastndashalgebra C Thisis the smallest set such that for all m ge 1

1 983042Q(x0 xmminus1)983042 isin F(C ) for all lowast-polynomials innon-commuting algebras with coefficients in C

2 If ϕj isin F(C ) for j lt m and f Rm rarr R is continuous thenf (ϕ0 ϕmminus1) isin F(C )

3 If ϕ isin F(C ) and j isin N then sup983042xj983042le1 ϕ isin F(C ) andinf983042xj983042le1 ϕ isin F(C ) (For brevity wersquoll write supxj and infxj and wersquoll use x y z for variables)

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Examples

1 ϕab(x y) = 983042xy minus yx983042 supx 983042xy minus yx983042 supx y 983042xy minus yx9830422 ϕproj(x) = max(983042xlowast minus x983042 983042x2 minus x983042)3 ϕO2(x y) = max(983042xlowastx minus 1983042 983042ylowasty minus 1983042 983042xxlowast + yylowast minus 1983042)

If C le A then the interpretation of ϕ(x0 xkminus1) isin F(C ) in A ata0 akminus1 is defined recursively in the natural way and denoted

ϕ(a0 akminus1)A

ConventionIrsquoll write x for x0 xkminus1 a for a0 akminus1 for some k ge 1lsquoclear from the contextrsquo

LemmaThe evaluation Ak rarr R a 983041rarr ϕ(a)A is uniformly continuous onevery bounded ball The image of every bounded ball under theevaluation map is a bounded subset of R

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

The fundamental theorem of ultraproducts

Theorem (983040Los)

For every formula ϕ(x) isin F(A) and for every a isin983124

U An we have

ϕ(a)983124

U A = limnrarrU

ϕ(an)An

We proved this two slides ago before we defined formulas Letrsquosdo it again

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Proof of 983040Losrsquos Theorem the case of AUBy induction on complexity of ϕ prove that for all ϕ and a

ϕ(a)AU = limnrarrU

ϕ(an)A (1)

(a) If ϕ(a) = 983042Q(a)983042 then by the definition of cU (A)

983042Q(a)983042 = limnrarrU 983042Q(an)983042

(b) If (1) is true for ϕ0 ϕkminus1 and f Rk rarr R is continuousthen

f (ϕ0(a) ϕkminus1(a)983042))AU = f (ϕ0(a)AU ϕkminus1(a)983042)AU )

= f ( limnrarrU

ϕ0(an)A lim

nrarrUϕkminus1(an)983042)A)

= limnrarrU

f (ϕ0(an) ϕkminus1(an)983042)A

(c) If (1) is true for ϕ(a x) then

supxisinAU 983042x983042le1 ϕ(a x) = limnrarrU supxisinA983042x983042le1 ϕ(a x)

The proof for infx is analogous

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Consequences of 983040Losrsquos Theorem

1 A is abelian iff AU is(supx y 983042xy minus yx983042)A = (supx y 983042xy minus yx983042)AU

2 A is purely infinite and simple iff AU is (Usesupx supy min(max(983042xlowastx983042 minus 12 0) infz 9830422zxlowastxzlowast minus ylowasty983042))

3 A has real rank zero iff AU does (lsquoEvery self-adjointcontraction is within 1n of a linear combination of 2n + 2commuting projectionslsquo)

4 A is n-subhomogeneous iff AU is

5 A has stable rank one iff AU does

6 A has strict comparison of positive elements by quasitracesiff AU does

7 A and AU have the same radius of comparison

and so on (See lsquoModel theory of Clowastndashalgebras I Farah BHart M Lupini L Robert A Tikuisis A Vignati W WinterMemoirs AMS to appear)

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Other categories

In most other categories equipped with an lsquoultrapower functorrsquo ananalog of 983040Losrsquos Theorem holdsUltraproducts of Cuntz semigroups were defined byAntoinendashPererandashThiel1

QuestionIs there a lsquo983040Los Theoremrsquo for ultrapowers of Cuntz semigroups Of

course I am asking whether Cuntz semigroups are axiomatizable in some

reasonable logic

1Cuntz semigroups will not be mentioned again in these talks if neitherW (A) nor Cu(A) rings a bell donrsquot worry

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Elementary submodels

For n ge 0 let Fn(C ) be the set of all formulas in F(C ) with freevariables included in x0 xnminus1If A le B we say A is an elementary submodel of B (A ≼ B) if

ϕ(a)A = ϕ(a)B

for all n all ϕ isin Fn(C) and all a in An1 (Equivalently we can ask

that this holds for all ϕ isin Fn(A))

DefinitionThe diagonal embedding ι A rarr AF is ι(a) = (a a a )We routinely identify A with ι[A]

Corollary (983040Los)

A ≼ AU

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

Expanding the language

Suppose Φ A rarr A is a contraction We can expand the languageby replacing lowast-polynomials Q(x) with expressions that in additioninvolve Φ For example if

ϕ = supx y

983042Φ(x + y)minus Φ(x)minus Φ(y)983042

then ϕ(AΦ) = 0 iff Φ(a+ b) = Φ(a) + Φ(b) for all a b in AWe can similarly expand the language by adding symbols forstates etc

Theorem (983040Los)

(AΦ) ≺ (AU ΦU )

In particular Φ isin Aut(A) iff ΦU isin Aut(AU )

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

The analog of 983040Losrsquos Theorem for AF

LemmaFn(C) is an R-algebra with the seminorm

983042ϕ(x)983042 = supAa

ϕ(a)A

where the sup is taken over all A and all n-tuples a in A1

DefinitionThe theory of A is the evaluation functional Th(A) F0(C) rarr R

ϕ 983041rarr ϕA

Th(A) is a character hence determined by its kernel Someauthors define the theory of A to be the kernel of Th(A)

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers

The analog of 983040Losrsquos Theorem for AF

983040Los says Th(A) = Th(AU )

Theorem (Ghasemi)

Th(Ainfin) can be computed from Th(A)More generally if the restriction of F to every F-positive set is notan ultrafilter then Th(AF ) depends only on Th(A)

Corollary

For every ATh(Ainfin) = Th((Ainfin)infin) = Th(AZdensity zero

) = Th(AIsummable)

In the next lecture we will now consider the other importantproperty of ultrapowers