Introduction to Quantum Groups Operator algebraic aspects

Introduction to Quantum Groups Operator algebraic aspects

Roland Vergnioux

University of Normandy (France)

Herstmonceux, 15 July 2015

Outline

1 Constructions and tools

[Woronowicz, Baaj–Skandalis, Banica, ...]

– Full and reducedC^∗-algebra – The dualC^∗-algebra – Corepresentations – The boundary ofFO(Q)

2 Some operator-algebraic properties

[Voiculescu, Ruan, Tomatsu, Brannan, DFSW, DCFY]

– Amenability

– Approximation properties – Haagerup AP forFO(Q)

The algebraic Setting

General setting

We are given: A unital∗-algebra and ∆ :A →A A ∗-hom.

Assume: A generated by uij such that∆(uij) =P

uik⊗u_kj, u = (u_ij) andu¯= (u_ij^∗) invertible in M_N(A).

I(id⊗∆)∆ = (∆⊗id)∆ and

ISpan∆(A)(1A) =Span∆(A)(A 1) =A A Examples

G ⊂U_N compact group of matrices

A =Pol(G), uij :G →C coordinate maps

∆(u_ij)(g,h) =P

g_ikh_kj =u_ij(gh) Γ =hγ_iifinitely generated group A =C[Γ],u =diag(γ_i), ∆(γ) =γ⊗γ

The algebraic Setting

General setting

We are given: A unital∗-algebra and ∆ :A →A A ∗-hom.

Assume: A generated by uij such that∆(uij) =P

uik⊗u_kj, u = (u_ij) andu¯= (u_ij^∗) invertible in M_N(A).

I(id⊗∆)∆ = (∆⊗id)∆ and

ISpan∆(A)(1A) =Span∆(A)(A 1) =A A Notation: A =Pol(G) =C[Γ].

G,Γ are a “compact and a discrete quantum group in duality”.

The algebraic Setting

General setting

We are given: A unital∗-algebra and ∆ :A →A A ∗-hom.

Assume: A generated by uij such that∆(uij) =P

uik⊗u_kj, u = (u_ij) andu¯= (u_ij^∗) invertible in M_N(A).

I(id⊗∆)∆ = (∆⊗id)∆ and

ISpan∆(A)(1A) =Span∆(A)(A 1) =A A Examples

A =Ao(Q), Q ∈GL_N(C): A_o(Q) =hu_ij |u unitary,u =Q¯uQ⁻¹i The elementsUij =P

uik⊗u_kj satisfy the relations I∆.

Notation: Γ=FO(Q),G=O⁺(Q).

Special case Q =

0 1

−1/q0

IAo(Q) =Pol(SUq(2))

The full and reduced C

-algebras

Definition (Full C^∗-algebra) We wish to define a norm on A by

kxk_f = sup{kπ(x)k_B(K) |π:A →B(K) ∗-rep}.

Assumption: this is finite — “A has an envelopping C^∗-algebra”.

IcompletionA_f =C_f(G) =C_f^∗(Γ)

By universality, the coproduct ∆ extends to

∆ :C_f^∗(Γ)→C_f^∗(Γ)⊗C_f^∗(Γ)

and C_f^∗(Γ) is a Woronowicz C^∗-algebra (cf Adam’s talk).

The full and reduced C

-algebras

Still assume: A has an enveloppingC^∗-algebra — e.g. (uij) unitary.

Theorem: there exists a unique stateh :A →Csuch that (h⊗id)∆ = (id⊗h)∆ = 1◦h, and it is faithful on A. Definition (Reduced C^∗-algebra)

(A,h) IΛ :A ,→H=L²(G) =`²(Γ) with kΛ(x)k²=h(x^∗x)

Iλ:A →B(H),λ(x)Λ(y) = Λ(xy)

IcompletionA_r=C_r(G) =C_r^∗(Γ) =Imgλ

By invariance, the coproduct ∆ extends to

∆ :C_r^∗(Γ)→C_r^∗(Γ)⊗C_r^∗(Γ)

and C_r^∗(Γ) is a Woronowicz C^∗-algebra (cf Adam’s talk).

The full and reduced C

-algebras

Still assume: C[Γ] has an envelopping C^∗-algebra — e.g. (uij) unitary.

Ipotentially differentWoronowicz C^∗-algebras C_f^∗(Γ),C_r^∗(Γ)

Ipossibly others in between Useful maps:

λ:C_f^∗(Γ)→C_r^∗(Γ) by definition (regular representation) :C_f^∗(Γ)→Ccharacter s.t. (u_ij) =δ_ij (trivial repr. / co-unit)

∆⁰:C_r^∗(Γ)→C_r^∗(Γ)⊗C_f^∗(Γ) (Fell’s absorption)

The dual C

-algebra

Multiplicative Unitary

DefineV ∈B(H⊗H) by putting V(Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(∆(x)(1⊗y)).

IC_r^∗(Γ) =Span{(ω⊗id)(V)|ω ∈B(H)∗}

IV(x⊗1)V^∗= ∆(x) forx ∈C_r^∗(Γ)

The dual C

-algebra

Multiplicative Unitary

DefineV ∈B(H⊗H) by putting V(Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(∆(x)(1⊗y)).

Definition (Dual C^∗-algebra)

We define ˆA=C0(Γ) =C^∗(G) and a coproduct ∆ : ˆA→M(ˆA⊗A) byˆ C₀(Γ) =Span{(id⊗ω)(V)|ω∈B(H)∗},

∆(f) =V^∗(1⊗f)V for f ∈C₀(Γ).

We put also C_b(Γ) =M(C0(Γ)).

The dual C

-algebra

Definition (Dual C^∗-algebra)

We define ˆA=C0(Γ) =C^∗(G) and a coproduct ∆ : ˆA→M(ˆA⊗A) byˆ C₀(Γ) =Span{(id⊗ω)(V)|ω∈B(H)∗},

∆(f) =V^∗(1⊗f)V for f ∈C0(Γ).

We put also C_b(Γ) =M(C₀(Γ)).

Duality

We have V ∈M(C₀(Γ)⊗C_r^∗(Γ)) Iduality between C_r^∗(Γ) andC₀(Γ):

ω∈C_r^∗(Γ)^∗ Iω˜ = (id⊗ω)(V)∈C_b(Γ).

V lifts toV_f ∈M(C₀(Γ)⊗C_f^∗(Γ)) such thatV = (id⊗λ)(V_f), again:

ω∈C_f^∗(Γ)^∗ Iω˜ = (id⊗ω)(V_f)∈C_b(Γ).

Irreducible (co)representations

Definition

K f.-d. Hilbert space,v ∈B(K)⊗C_f^∗(Γ) unitary, (id⊗∆)(v) =v12v13 I“representation” ofG/ “corepresentation” ofΓ andA

Following the theory of the compact case:

f ∈B(K₁,K₂) intertwiner if (f⊗1)v₁=v₂(f⊗1) IHom(v₁,v₂) v₁∼v₂ ifHom(v₁,v₂) contains a bijection

v irreducible if Hom(v,v) =Cid Iset IrrΓ

direct sum, tensor product, conjugate representation, ...

Irreducible (co)representations

Definition

K f.-d. Hilbert space,v ∈B(K)⊗C_f^∗(Γ) unitary, (id⊗∆)(v) =v12v13 I“representation” ofG/ “corepresentation” ofΓ andA

Application: “Decomposition of the regular repr. ofG”

There is an isomorphism C₀(Γ)'L

α∈IrrΓB(K_α) s.t. V_f 'L v_α. Moreover (∆⊗id)(V_f) =Vf,13Vf,23'L

vα⊗v_β.

Remark: in fact f.d. representations v live in B(K)A.

Iallows to reconstructA from a WoronowiczC^∗-algebra

Corepresentations of F O(Q) and the boundary

In this slide Γ=FO(Q) andQQ¯ ∈CIN. Theorem

One can write IrrFO(Q) ={v_k} with v0 = 1, v1 =u,v¯_k 'v_k and v_k⊗v₁ 'vk−1⊕v_k+1.

Corepresentations of F O(Q) and the boundary

In this slide Γ=FO(Q) andQQ¯ ∈CIN. Theorem

One can write IrrFO(Q) ={v_k} with v0 = 1, v1 =u,v¯_k 'v_k and v_k⊗v₁ 'vk−1⊕v_k+1.

An application

r ∈Hom(v_k+1,v_k⊗v₁) isometric IUCP map (cf Mike’s talk) R = Φ^k,1_k+1∗

:B(H_k)→B(H_k+1),f 7→r^∗(f⊗id)r.

We putC(∂FO(Q)) = lim−→(B(H_k),R).

Recall B(Hk)⊂C0(Γ) I∂Γ= “projective limit of spheres”.

Corepresentations of F O(Q) and the boundary

In this slide Γ=FO(Q) andQQ¯ ∈CIN. An application

r ∈Hom(v_k+1,v_k⊗v₁) isometric IUCP map (cf Mike’s talk) R = Φ^k,1_k+1∗

:B(Hk)→B(Hk+1),f 7→r^∗(f⊗id)r.

We putC(∂FO(Q)) = lim−→(B(Hk),R).

Recall B(H_k)⊂C₀(Γ) I∂Γ= “projective limit of spheres”.

Theorem

C(∂Γ)⊂C_b(Γ)/C₀(Γ) is a infinite-dimensional unital∗-subalgebra.

It is stable under the left and right actions of Γ.

The restriction of the left (resp. right) action is amenable (resp. trivial).

Application: exactness and property “AO⁺”, solidity of the von Neumann

Amenability : definition(s)

Definition Γ is called

weakly amenable if C_b(Γ) admits an invariant state m:

∀ ω∈Cb(Γ)∗,f ∈Cb(Γ) m((ω⊗id)∆(f)) =ω(1)m(f).

strongly amenable if λ:C_f^∗(Γ)→C_r^∗(Γ) is an isomorphism.

Note: strongly amenable ⇔ factors through λ

⇔ almost invariant vectors in H.

For Γ classical, m:P(Γ)→[0,1] invariant, finitely additive,m(Γ) = 1.

Theorem

For discrete quantum groups, weakly amenable ⇔ strongly amenable.

“⇒” is harder ifh is not tracial, and still open in the locally compact case.

Amenability : examples

Fusion rules: α⊗β 'L

m^γ_αβγ with α,β,γ ∈IrrΓ Theorem

Assume Γ,Λhave the same fusion rules. If Γ is amenable, dimv_α^Γ≤dimv_α^Λfor all α. Λis amenableiff we have =for all α.

“Amenability is a property of the dimension function on the fusion ring.”

Examples

Finite or abelian groups are amenable ; non-ab. free groups are not.

The dual of a classicalG is always amenable.

The dual ofSU_q(2) is amenable.

FO(Q) is amenableiffN = 2.

∗

Approximation properties

Fix (A,h) unital separable C^∗-algebra with faithful state.

Approximation property: ∃Tn:A→As.t. ∀a∈A kT_n(a)−ak −→

n∞ 0.

Some examples:

CPAP: T_n UCP and finite rank

CBAP: Tn uniformly CB and finite rank HAP:Tn UCP and compact onL²(A,h) Theorem

Γ amenable⇒ C_r^∗(Γ) has the CPAP. ⇐holds if h is tracial.

Proof. ⇒: T_ϕ = (id⊗ϕ)◦∆⁰ for ϕ∈C_f^∗(Γ)^∗,T=id.

Strong amenability ⇔ approximated by vector states for λ

⇔ by statesϕsuch that ˜ϕhas finite rank.

⇐: have to reconstructϕfrom T.

Approximation properties

Fix (A,h) unital separable C^∗-algebra with faithful state.

Approximation property: ∃Tn:A→As.t. ∀a∈A kT_n(a)−ak −→

n∞ 0.

Some examples:

CPAP: T_n UCP and finite rank

CBAP: Tn uniformly CB and finite rank HAP:Tn UCP and compact onL²(A,h) Theorem

If there exist states ϕn∈C_f^∗(Γ)^∗ s.t. ϕn→∗-weakly and ϕ˜n∈C0(Γ), then C_r^∗(Γ) has the HAP.⇐ holds if h is a trace.

Proof. ⇒: Tϕ = (id⊗ϕ)◦∆⁰ for ϕ∈C_f^∗(Γ)^∗,T=id.

⇐: have to reconstructϕfrom T.

F O(Q) has the HAP

Considerϕt given by ˜ϕt =X_[k+1]t

[k+1]N idk ∈L

B(Hk).

Clearly ˜ϕt∈C0(Γ) andϕt →as t→N. Isϕt a state?

Approach 1 (Q =I_N) Uses B =hP

u_iii ⊂C_f^∗(Γ).

In the unimodular case, the “orthogonal projection” extends to a positive contraction P :C_f^∗(Γ)B.

For FO(I_N),B'C([−N,N]) and a computation shows thatϕ_t =ev_t◦P.

F O(Q) has the HAP

Considerϕt given by ˜ϕt =X_[k+1]t

[k+1]N idk ∈L

B(Hk).

Clearly ˜ϕt∈C0(Γ) andϕt →as t→N. Isϕt a state?

Monoidal equivalence

“Abstract” equivalence F :CorepΓ₁ →CorepΓ₂ (⇒ same fusion rules).

Classical cases G, Γ: implies isomorphism.

Every O⁺(Q) is monoidally equivalent to anSU_q(2).

Approach 2 If ˜ϕ=P

f(α)idα andΓ∼_monΓ⁰, defineϕ⁰ onC_f^∗(Γ⁰) by ˜ϕ⁰ =P

f(α)id⁰_α. Fact: kT_ϕk_cb=kT_ϕ⁰k_cb. In particular ϕstate⇔ ϕ⁰ state.

Proposition

On C(SU (2)),ϕ is the vaccum state in the “Podle´s sphere”