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 ▶ dual Hopf C^{∗}-algebras S, Ŝ ▶ regular representations λ : S ↠ S_{red} ⊂ L(H)

and Ŝ → Ŝ_{red} ⊂ L(H) ▶ trivial representations ε : S → ℂ, ε̂ : Ŝ → ℂ

Woronowicz C^{∗}-algebras ▶ S unital Hopf-C^{∗}-algebra with

δ(S)(1⊗S) and δ(S)(S ⊗1) dense in S ⊗S ▶ category of corepresentations C s.t.

Ŝ = Ŝ_{red} = ⨁_{r∈Irr C} p_{r}Ŝ and p_{r}Ŝ ≃ L(H_{r}) Ex. : S = S_{red} = C(G), H = ℓ^{2}(G) Ex. : S = C^{∗}(Γ), Irr C = Γ, p_{r} = 11_{{r}} ∈ C_{0}(Γ)

Crossed products If S_{red} coacts on A, one can define crossedproduct C^{∗}-algebras A ⋊ Ŝ, A ⋊_{red} Ŝ.

Descent Morphisms

▶ S, Ŝ : full C^{∗}-alg. of a l.c. quantum group ▶ A, B endowed with coactions of S_{red} Proposition We put E ⋊ Ŝ = E ⊗_{B}(B ⋊ Ŝ)

1. K_{B⋊Ŝ}(E ⋊ Ŝ) ≃ K_{B}(E) ⋊ Ŝ.

2. If (E, π, F) ∈ 𝔼_{S_{red}}(A, B) then (E ⋊ Ŝ, π ⋊ id, F ⋊ 1) ∈ 𝔼(A ⋊ Ŝ, B ⋊ Ŝ).

3. this defines a morphism j from KK_{S_{red}}(A, B) to KK(A ⋊ Ŝ, B ⋊ Ŝ)

4. j(x⊗_{D}y) = j(x)⊗_{D⋊Ŝ} j(y)

Green-Julg Theorem

▶ S_{red} : Woronowicz C^{∗}-algebra ▶ A, B endowed with coactions δ_{A}, δ_{B} of S_{red} ▶ δ_{A} trivial, B S_{red}-algebra

Theorem There is an isomorphism KK_{S_{red}}(A, B) ≃ KK(A, B ⋊_{red} Ŝ) given by

KK_{S_{red}}(A, B)
j_{r}

KK(A ⋊_{red} Ŝ, B ⋊_{red} Ŝ)

ϕ^{∗} KK_{S_{red}}(A_{1}, (B ⋊_{red} Ŝ)_{1})

· ⊗β

KK(A, B ⋊_{red} Ŝ)

ψ

K-amenability

▶ S, S_{red}, Ŝ, Ŝ_{red} : C^{∗}-algebras of a locally

compact quantum group ▶ λ : S → S_{red} : regular representation ▶ ε : S → ℂ : trivial representation Ex. : S = C^{∗}(G), Ŝ = Ŝ_{red} = C_{0}(G)

Theorem We have i ⇒ ii ⇒ iii ⇒ iv, and iv ⇒ i when S_{red} is unital (discrete case) : i. 11 ∈ KKŜ_{red}(ℂ, ℂ) is represented by (E, 1, F)

with δ_{E} ≺ δŜ_{red} (K-amenability) ii. ∀ A [λ_{A}] ∈ KK(A⋊S, A⋊_{red} S) is invertible iii. [λ] ∈ KK(S, S_{red}) is invertible iv. ∃ α ∈ KK(S_{red}, ℂ) λ^{∗}(α) = [ε] ∈ KK(S, ℂ).

Amalgamated Free Products

▶ T ⊂ S_{1}, S_{2} : amenable Wor. C^{∗}-algebras ▶ S = S_{1} ∗_{T} S_{2} : amalgamated free product ▶ P : S ↠ T, R_{i} : S ↠ S_{i} cond. expect. ▶ E, F_{i} associated GNS constructions ▶ ε : T, S_{i} → ℂ trivial representation Ex. : S_{i} = C^{∗}(Γ_{i}), T = C^{∗}(Δ), Γ = Γ_{1} ∗_{Δ} Γ_{2}

Definition Quantum Serre Tree ▶ H = F_{1} ⊗_{ε}ℂ ⊕ F_{2} ⊗_{ε}ℂ, K_{0} = E ⊗_{ε}ℂ ▶ GNS representations of S The classical « Serre » tree (V, E) associated to « Irr C_{1} ∗_{D} Irr C_{2} » induces a decomposition H = ⨁ p_{n}H and the J.-V. operator F :

▶ E(r, i)^{∘}⊗_{ε}ℂ →^{∼} F_{i}^{∘}⊗_{ε}ℂ, ηℂ → η_{2} ⊗_{ε}1_{ℂ}

Theorem

1. (H ⊕ K_{0}, π_{GNS} , F) defines γ ∈ KK(S_{red}, ℂ)

2. (Ŝ, δ̂) is K-amenable

Quantum Cayley Graph

▶ S Wor. C^{∗}-algebra, Ŝ = Ŝ_{red} ⊂ L(H) ▶ p_{r} ∈ Ŝ min. central proj. (r ∈ Irr C) ▶ p_{1} = ∑_{r∈D} p_{r}, 1_{C} ∉ D, D̄ = D Ex. : S = C^{∗}(Γ), Ŝ = C_{0}(Γ), p_{r} = 11_{{r}}, r ∈ Γ

Definition Quantum Cayley Graph ▶ H : space of the regular repr. of S, Ŝ ▶ K = H ⊗p_{1}H, S = id⊗ϵ : K → H ▶ Θ = Σ(1⊗U)V (U ⊗U)Σ ▶ regular repr. on H, trivial repr. on p_{1}H

Definition Classical Cayley Graph ▶ V = Irr C, θ(r, r′) = (r′, r) ▶ E = {(r, r′) ∈ V^{2} | r′ ⊂ r⊗D} ▶ C_{0}(V) → L(H), 11_{{r}} ↦ 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 p_{⋆+} ∈ L(H) « on ascending edges ».

Problems ^{▶} Θ^{2} ≠ 1 ▶ p_{+⋆} := 1−Θp_{⋆+} Θ^{∗} ≠ p_{⋆+} ^{▶} p_{++} = p_{⋆+} p_{+⋆} ▶ [p_{⋆+} , u_{i,j}] is compact but not of finite rank ▶ F = T p_{++} : K_{g} → H is not Fredholm

Theorem There exists a natural representation π_{∞} : A_{o}(Q) → L(H_{∞}) such that

K_{g p++}

K_{++}^B H

H_∞^{R∗}

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