KK-theory for quantum groups : functorial and geometrical methods

KK -theory for quantum groups : functorial and

geometrical methods

Roland Vergnioux vergniou@math.jussieu.fr

May 15, 2003

Crossed Products by Quantum Groups L.c. quantum groups

I dual Hopf C^∗-algebras S, Sˆ

I regular representations λ : S _{Sred ⊂ L(H)} and Sˆ → Sˆ_red ⊂ L(H)

I trivial representations ε : S → C_{, ˆε : ˆS →} C

Woronowicz C^∗-algebras

I S unital Hopf-C^∗-algebra with

δ(S)(1⊗S) and δ(S)(S⊗1) dense in S⊗S

I category of corepresentations C s.t.

Sˆ = ˆS_red = ^L_r∈IrrC p_rSˆ and p_rSˆ ' L(H_r) Ex. : S = S_red = C(G), H = `²(G)

Ex. : S = C^∗(Γ), Irr C = Γ, p_r = 11_{r} ∈ C₀(Γ)

Crossed products

If S_red coacts on A, one can define crossed- product C^∗-algebras A o Sˆ, A o_red Sˆ.

Descent Morphisms

I S, Sˆ : full C^∗-alg. of a l.c. quantum group

I A, B endowed with coactions of S_red

Proposition

We put E o Sˆ = E⊗_B(B o Sˆ) 1. K_BoSˆ(E o _{Sˆ) ' KB(E)} o _Sˆ. 2. If (E, π, F) ∈ E

S_red(A, B) then

(E o _{S, πˆ} o _{id, F} o _{1) ∈} E_(A o _{S, Bˆ} o _Sˆ).

3. this defines a morphism j from KK_Sred(A, B) to KK(A o S, Bˆ o Sˆ)

4. j(x⊗_Dy) = j(x)⊗_DoSˆ j(y)

Green-Julg Theorem

I S_red : Woronowicz C^∗-algebra

I A, B endowed with coactions δ_A, δ_B of S_red

I δ_A trivial, B S_red-algebra

Theorem There is an isomorphism

KK_Sred(A, B) ' KK(A, B o_red Sˆ) given by

KK_Sred(A, B) j_r

KK(A o_red S, Bˆ o_red Sˆ)

φ^∗ KK_Sred(A₁,(B o_red Sˆ)₁)

· ⊗β

KK(A, B o_red Sˆ) ψ

K-amenability

I S, S_red, Sˆ, Sˆ_red : C^∗-algebras of a locally compact quantum group

I λ : S → S_red : regular representation

I ε : S → C : trivial representation Ex. : S = C^∗(G), Sˆ = ˆS_red = C₀(G)

Theorem We have i ⇒ ii ⇒ iii ⇒ iv, and iv ⇒ i when S_red is unital (discrete case) :

i. 11 ∈ KK_Sˆ

red(C_,C₎ is represented by (E,1, F) with δ_E ≺ δ_Sˆ

red (K-amenability)

ii. ∀ A [λ_A] ∈ KK(Ao_{S, A}o_redS) is invertible iii. [λ] ∈ KK(S, S_red) is invertible

iv. ∃ α ∈ KK(S ,C_{) λ}^∗_{(α) = [ε] ∈ KK(S,}C_).

Amalgamated Free Products

I T ⊂ S₁, S₂ : amenable Wor. C^∗-algebras

I S = S₁ ∗_T S₂ : amalgamated free product

I P : S _{T, Ri : S Si} cond. expect.

I E, F_i associated GNS constructions

I ε : T, S_i → C trivial representation

Ex. : S_i = C^∗(Γ_i), T = C^∗(∆), Γ = Γ₁ ∗_∆ Γ₂

Definition Quantum Serre Tree

I H = F₁⊗_εC _{⊕ F2⊗ε}C_{, K0 = E⊗ε}C

I GNS representations of S

The classical « Serre » tree (V, E) associated to « Irr C₁ ∗_D Irr C₂ » induces a decomposition H = ^Lp_nH and the J.-V. operator F :

I E(r, i)^◦⊗_εC _→^∼ _F^◦

i ⊗_εC_{, η}C _{→ η2⊗ε1C}

Theorem

1. (H ⊕ K₀, π_GNS, F) defines γ ∈ KK(S_red,C₎ 2. ( ˆS,ˆδ) is K-amenable

Quantum Cayley Graph

I S Wor. C^∗-algebra, Sˆ = ˆS_red ⊂ L(H)

I p_r ∈ Sˆ min. central proj. (r ∈ Irr C)

I p₁ = ^P_r∈D p_r, 1_C ∈ D/ , D¯ = D

Ex. : S = C^∗(Γ), Sˆ = C₀(Γ), p_r = 11_{r}, r ∈ Γ

Definition Quantum Cayley Graph

I H : space of the regular repr. of S, Sˆ

I K = H⊗p₁H, S = id⊗ : K → H

I Θ = Σ(1⊗U)V (U ⊗U)Σ

I regular repr. on H, trivial repr. on p₁H

Definition Classical Cayley Graph

I V = Irr C, θ(r, r⁰) = (r⁰, r)

I E = {(r, r⁰) ∈ V ² | r⁰ ⊂ r⊗D}

I C₀(V ) → L(H), 11_{r} 7→ p_r

Free Quantum Groups

(E, V ) is a tree iff S is a free product of free quantum groups A_o(Q), A_u(Q) (Wang, Banica).

Then (E, V ) induces a projection pF₊ ∈ L(H)

« on ascending edges ».

Problems ^I Θ² 6= 1

I p+F := 1−ΘpF₊Θ^∗ 6= pF₊ I p++ = pF₊p+F

I [pF₊, u_i,j] is compact but not of finite rank

I F = T p++ : K_g → H is not Fredholm

Theorem There exists a natural representa- tion π_∞ : A_o(Q) → L(H_∞) such that

K_{g p}

++

K++ B

H H_∞ ^R∗

defines γ ∈ KK_Sˆ(C_,C_{) when Tr Q}^∗_{Q > 2. In} the classical case H_∞ = 0.