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
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