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?

NO!

f(C)N (θ) = 1

N

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

Note:

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

1

N·

(sin(N2 θ)

sin( 12θ)

)2

=N∑

n=−N(1− |n|

N) fn e

inθ

Kenny De Commer (VUB) Approximation properties 15 July 2014 2 / 14

Approximation properties

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly? NO!

f(C)N (θ) = 1

N

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

Note:

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

1

N·

(sin(N2 θ)

sin( 12θ)

)2

=N∑

n=−N(1− |n|

N) fn e

inθ

Kenny De Commer (VUB) Approximation properties 15 July 2014 2 / 14

Approximation properties

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly? NO!

f(C)N (θ) = 1

N

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

Note:

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

1

N·

(sin(N2 θ)

sin( 12θ)

)2

=N∑

n=−N(1− |n|

N) fn e

inθ

Kenny De Commer (VUB) Approximation properties 15 July 2014 2 / 14

Approximation properties

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly? NO!

f(C)N (θ) = 1

N

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

Note:

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

1

N·

(sin(N2 θ)

sin( 12θ)

)2

=N∑

n=−N(1− |n|

N) fn e

inθ

Kenny De Commer (VUB) Approximation properties 15 July 2014 2 / 14

Approximation properties

Fourier series

Let f ∈ C (S1).

fN(θ) =N∑−N

fn e inθ → f (θ) uniformly? NO!

f(C)N (θ) = 1

N

N−1∑n=0

fn(θ)→ f (θ) uniformly? Yes!

Note:

f(C)N (θ) = (gN ∗ f )(θ), gN(θ) =

1

N·

(sin(N2 θ)

sin( 12θ)

)2

=N∑

n=−N(1− |n|

N) fn e

inθ

Kenny De Commer (VUB) Approximation properties 15 July 2014 2 / 14

Approximation properties

Approximation properties

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 3 / 14

Approximation properties

Approximation properties

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 3 / 14

Approximation properties

Approximation properties

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 3 / 14

Approximation properties

Holomorphicity trick

(0, 1) ⊆ S ⊆ C

Lemma

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

If

θ =∑

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

then θ =∑

n θn in cb-norm.

Kenny De Commer (VUB) Approximation properties 15 July 2014 4 / 14

Approximation properties

Holomorphicity trick

(0, 1) ⊆ S ⊆ C

Lemma

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

If

θ =∑

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

then θ =∑

n θn in cb-norm.

Kenny De Commer (VUB) Approximation properties 15 July 2014 4 / 14

Approximation properties

Holomorphicity trick

(0, 1) ⊆ S ⊆ C

Lemma

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

If

θ =∑

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

then θ =∑

n θn in cb-norm.

Kenny De Commer (VUB) Approximation properties 15 July 2014 4 / 14

Compact quantum groups

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 5 / 14

Compact quantum groups

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 5 / 14

Compact quantum groups

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).

Kenny De Commer (VUB) Approximation properties 15 July 2014 6 / 14

Compact quantum groups

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).

Kenny De Commer (VUB) Approximation properties 15 July 2014 6 / 14

Compact quantum groups

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).

Kenny De Commer (VUB) Approximation properties 15 July 2014 6 / 14

Compact quantum groups

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).

Kenny De Commer (VUB) Approximation properties 15 July 2014 6 / 14

Compact quantum groups

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)

Kenny De Commer (VUB) Approximation properties 15 July 2014 7 / 14

Compact quantum groups

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)

Kenny De Commer (VUB) Approximation properties 15 July 2014 7 / 14

Compact quantum groups

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 8 / 14

Compact quantum groups

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 8 / 14

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 )

Kenny De Commer (VUB) Approximation properties 15 July 2014 9 / 14

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 )

Kenny De Commer (VUB) Approximation properties 15 July 2014 9 / 14

On the proof

Multipliers

Definition

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

θ(uπij ) = θπuπij , θπ ∈ C.

Lemma

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 10 / 14

On the proof

Multipliers

Definition

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

θ(uπij ) = θπuπij , θπ ∈ C.

Lemma

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 10 / 14

On the proof

Multipliers

Definition

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

θ(uπij ) = θπuπij , θπ ∈ C.

Lemma

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 10 / 14

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 11 / 14

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),

so

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 =

∑n

ω3n,zp(n/2)

satisfies conditions for holomorphicity trick.

Kenny De Commer (VUB) Approximation properties 15 July 2014 12 / 14

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic?

X

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 13 / 14

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic? X

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 13 / 14

On the proof

Details I

θ : S→ CB(SUq(2)) and holomorphic? X

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.

Kenny De Commer (VUB) Approximation properties 15 July 2014 13 / 14

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?

X

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

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

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?

X

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

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

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?

X

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

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

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? X

‖p(n/2)‖cb = O(|q|−2n),

|ωn,z |3 = O(|q|3n(1−<(z))).

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