Kenny De Commer - group30.ugent.be · Approximation properties for quantum groups Kenny De Commer...

Approximation properties for quantum groups

Kenny De Commer(joint work with A. Freslon and M. Yamashita)

Free University Brussels

15 July 2014

Kenny De Commer (VUB) Approximation properties 15 July 2014 1 / 14

Approximation properties

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly?

f(C)N (θ) = 1

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

(sin(N2 θ)

sin( 12θ)

n=−N(1− |n|

N) fn e

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly? NO!

f(C)N (θ) = 1

N−1∑n=0

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

(sin(N2 θ)

sin( 12θ)

n=−N(1− |n|

N) fn e

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

f(C)N (θ) = 1

N−1∑n=0

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

(sin(N2 θ)

sin( 12θ)

n=−N(1− |n|

N) fn e

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

f(C)N (θ) = 1

N−1∑n=0

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

(sin(N2 θ)

sin( 12θ)

n=−N(1− |n|

N) fn e

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

f(C)N (θ) = 1

N−1∑n=0

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

(sin(N2 θ)

sin( 12θ)

n=−N(1− |n|

N) fn e

A ⊆ B(H) a C∗-algebra.

T : A→ A ⇒ T (n) : Mn(A)→Mn(A)

• T cp (completely positive) ⇐⇒ all T (n) positive.

• T cb (completely bounded) ⇐⇒ ‖T‖cb = lim supn ‖T (n)‖ bounded.

Net θα : A→ A & ∀a ∈ A : θα(a)→ a

• A has CPAP ⇐⇒ all θα ∈ CP fr(A).

• A has ACPAP ⇐⇒ all θα ∈ CP(A) ∩ CCfr(A).

• A has CCAP ⇐⇒ all θα ∈ CCfr(A).

• A has CBAP ⇐⇒ all θα ∈ CB≤C ,fr(A), some C ≥ 1.

T : A→ A ⇒ T (n) : Mn(A)→Mn(A)

Net θα : A→ A & ∀a ∈ A : θα(a)→ a

T : A→ A ⇒ T (n) : Mn(A)→Mn(A)

Net θα : A→ A & ∀a ∈ A : θα(a)→ a

Holomorphicity trick

(0, 1) ⊆ S ⊆ C

Assume there exists holomorphic θ : S→ CB(A) such that

θt is cp for 0 < t < 1 and θt → id pointwise for t → 1.

For |z | < ε small, θz =∑

n θz,n in cb-norm with θz,n finite rankcb-map.

Then A has ACPAP.

Proof.

Consider S→ CB(A)/CBfr(A).

Remark

θ =∑

n θn pointwise on dense subspace and∑n ‖θn‖cb <∞

then θ =∑

n θn in cb-norm.

(0, 1) ⊆ S ⊆ C

Then A has ACPAP.

Proof.

Remark

θ =∑

then θ =∑

n θn in cb-norm.

(0, 1) ⊆ S ⊆ C

Then A has ACPAP.

Proof.

Remark

θ =∑

then θ =∑

n θn in cb-norm.

Compact quantum groups

Definition (Compact quantum group)

C (G) unital C∗-algebra, ∆ : C (G)→ C (G) ⊗min

C (G) coassociative and

∆(C (G))(C (G)⊗ 1) = ∆(C (G))(1⊗ C (G)) = C (G) ⊗min

C (G).

Example

C (G ), G compact group, ∆(f )(x , y) = f (xy)

C ∗u (Γ) and C ∗r (Γ), Γ discrete group, ∆(λg ) = λg ⊗ λg .

Remark

Change of viewpoint: C (G) = C ∗(Γ), with Γ = G discrete quantum group.

Definition (Compact quantum group)

C (G) unital C∗-algebra, ∆ : C (G)→ C (G) ⊗min

C (G) coassociative and

∆(C (G))(C (G)⊗ 1) = ∆(C (G))(1⊗ C (G)) = C (G) ⊗min

C (G).

Example

C (G ), G compact group, ∆(f )(x , y) = f (xy)

C ∗u (Γ) and C ∗r (Γ), Γ discrete group, ∆(λg ) = λg ⊗ λg .

Remark

Change of viewpoint: C (G) = C ∗(Γ), with Γ = G discrete quantum group.

Structure theory

P(G) ⊆ C (G) dense,

P(G) = C1⊕

Cuπ11 Cuπ12 . . . Cuπ1nCuπ21 Cuπ22 . . . Cuπ2n

......

. . ....

Cuπn1 Cuπn2 . . . Cuπnn

⊕ . . .

⇒ Haar state ϕ : C (G)→ C from P(G)→ C1.

⇒ Reduced C∗-algebra Cr (G) ⊆ B(L2(G)).

⇒ Universal C∗-algebra Cu(G).

Structure theory

P(G) = C1⊕

......

. . ....

⊕ . . .

Structure theory

P(G) = C1⊕

......

. . ....

⊕ . . .

Structure theory

P(G) = C1⊕

......

. . ....

⊕ . . .

Universal compact quantum groups

Example (Universal unitary and orthogonal quantum groups)

F ∈ Mn(C) invertible, U = (uij) indeterminates.

Cu(U+(F )) = C ∗u (FUF ) =< uij | U,FUF−1 unitary > .

Cu(O+(F )) = C ∗u (FOF ) =< uij | U = FUF−1 unitary > .

Remark

Cu(U+(In))/ < [uij , ukl ] >= C (Un)

Cu(O+(In))/ < [uij , ukl ] >= C (On)

C ∗u (FUIn)/ < uij − uji >= C ∗u (Fn)

Universal compact quantum groups

Example (Universal unitary and orthogonal quantum groups)

F ∈ Mn(C) invertible, U = (uij) indeterminates.

Cu(U+(F )) = C ∗u (FUF ) =< uij | U,FUF−1 unitary > .

Cu(O+(F )) = C ∗u (FOF ) =< uij | U = FUF−1 unitary > .

Remark

Cu(U+(In))/ < [uij , ukl ] >= C (Un)

Cu(O+(In))/ < [uij , ukl ] >= C (On)

C ∗u (FUIn)/ < uij − uji >= C ∗u (Fn)

Main theorem

Theorem (DC-Freslon-Yamashita, ‘13)

Let F ∈ Mn(C) invertible.

Then Cr (O+(F )) and Cr (U+(F )) have the ACPAP.

Remark

Earlier partial results for F = In by M. Brannan and A. Freslon.

Main theorem

Theorem (DC-Freslon-Yamashita, ‘13)

Let F ∈ Mn(C) invertible.

Then Cr (O+(F )) and Cr (U+(F )) have the ACPAP.

Remark

Earlier partial results for F = In by M. Brannan and A. Freslon.

On the proof

More on universal orthogonal quantum groups

Definition (Woronowicz)

Let 0 < |q| ≤ 1.

C (SUq(2)) =< α, β |{αβ = qβα, β∗β = ββ∗, α∗β = q−1βα∗

α∗α + β∗β = 1, αα∗ + q2ββ∗ = 1>

⇒ C (SUq(2)) = Cr=u(O+(F )) for F =

(0 −sgn(q)|q|1/2

|q|−1/2 0

Remark

Assume F F = ±1.

Fus(O+F ) = Fus(SU(2)).

Rep(O+F ) = Rep(SU∓q(2)) for q + q−1 = Tr(F ∗F )

On the proof

More on universal orthogonal quantum groups

Definition (Woronowicz)

Let 0 < |q| ≤ 1.

C (SUq(2)) =< α, β |{αβ = qβα, β∗β = ββ∗, α∗β = q−1βα∗

α∗α + β∗β = 1, αα∗ + q2ββ∗ = 1>

⇒ C (SUq(2)) = Cr=u(O+(F )) for F =

(0 −sgn(q)|q|1/2

|q|−1/2 0

Remark

Assume F F = ±1.

Fus(O+F ) = Fus(SU(2)).

Rep(O+F ) = Rep(SU∓q(2)) for q + q−1 = Tr(F ∗F )

On the proof

Multipliers

Definition

θ : C (G)→ C (G) is central multiplier if

θ(uπij ) = θπuπij , θπ ∈ C.

By ω → θω = (id⊗ ω)∆, there is a one-to-one correspondence between

Central positive functionals ω on Cu(G), i.e. ω(uπij ) = θπδij .

Completely positive central multipliers on Cr (G).

Remark

If ω ∈ Cu(G)∗, then θω is cb.

On the proof

Multipliers

Definition

Remark

On the proof

Multipliers

Definition

Remark

On the proof

Method of proof

1 Prove ACPAP for C (SUq(2)) with 0 < |q| < 1 by means of

θα completely positive central multipliersholomorphicity trick

2 Extend to Cr (O+(F )) with F F = ±1 by monoidal equivalence.

3 Extend to arbitrary Cr (O+(F )) and Cr (U+(F )) by free producttechniques.

On the proof

Explicit form

Take µn the nth dilated Chebyshev polynomial,

µn ∈ C[z ], µ0(z) = 1, zµn(z) = µn+1(z) + µn−1(z),

µn(x + x−1) =x−(n+1) − xn+1

x−1 − x.

Define

ωz(u(n/2)ij ) = ωn,zδij =

µn(|q|z + |q|−z)

µn(|q|+ |q|−1)δij .

Thenθz =

ω3n,zp(n/2)

satisfies conditions for holomorphicity trick.

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic?

Representation ρ : C (SUq(2))→ B(l2(N)) and

ωz(x) =〈ηz , ρ(x)ηz〉〈ηz , ηz〉

withS 3 z → ηz ∈ l2(N) holomorphic

Remark

Central states on Cu(G) ⇔ special states on C ∗u (DG).

Drinfeld double DSUq(2) = SLq(2,C).

Complementary series representations for SLq(2,C) with cyclic vector.

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic? X

Remark

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic? X

Remark

On the proof

Details II

θt is cp for 0 < t < 1? X

θt → id pointwise on P(SUq(2)) for t → 1? X∑n ‖θz,n‖cb <∞ for |z | < ε small?

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

On the proof

Details II

θt → id pointwise on P(SUq(2)) for t → 1? X

∑n ‖θz,n‖cb <∞ for |z | < ε small?

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

On the proof

Details II

θt → id pointwise on P(SUq(2)) for t → 1? X∑n ‖θz,n‖cb <∞ for |z | < ε small?

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

On the proof

Details II

θt → id pointwise on P(SUq(2)) for t → 1? X∑n ‖θz,n‖cb <∞ for |z | < ε small? X

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

Kenny De Commer - group30.ugent.be · Approximation properties for quantum groups Kenny De Commer...

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