KK-Theory for Quantum Groups : Functorial and Geometrical Methods

Roland Vergnioux October 25, 2003

1. Quantum Groups (notations)

2. Green-Julg Isomorphism

3. K-Amenability

4. Amalgamated Free Products

5. Quantum Cayley Graphs

6. Free Quantum Groups

7. Negative Type Function

Quantum Groups

Locally compact quantum groups ▶ “dual” Hopf C^{∗}-algebras S, Ŝ ▶ coproduct δ : S → M̃(S ⊗S) (coassociative) ▶ regular representations λ : S ↠ S_{red} ⊂ L(H)

and Ŝ → Ŝ_{red} ⊂ L(H) ▶ trivial representations ε : S → ℂ, ε̂ : Ŝ → ℂ Ex. : S = S_{red} = C_{0}(G), δ(f)(r, s) = f(rs),

Ŝ = C^{∗}(G), Ŝ_{red} = C_{red}^{∗}(G), H = L^{2}(G)

Discrete quantum groups ▶ S unital (Woronowicz C^{∗}-algebra) ▶ the category of corepr. C induces

Ŝ = Ŝ_{red} = ⨁_{r∈Irr C} p_{r}Ŝ with p_{r}Ŝ ≃ L(H_{r}) Ex. : S = C^{∗}(Γ), Ŝ = c_{0}(Γ), Irr C = Γ,

p_{r}Ŝ = ℂ11_{{r}} ∈ ℓ^{2}(Γ)

Green-Julg Theorem

▶ S unital (compact case) ▶ coactions δ_{A}, δ_{B} of S_{red} on A, B ▶ 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

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 Bass-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 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 Graphs

▶ S Wor. C^{∗}-algebra, p_{1} = ∑_{r∈D} p_{r} with ▶ D ⊂ Irr C finite, 1_{C} ∉ D, D̄ = D Ex. : S = C^{∗}(Γ), Ŝ = C_{0}(Γ), D ⊂ 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, K_{g} = Ker (Θ+id) ▶ 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′, s, i) ∈ V^{2} | r′ ⊂_{i} r⊗s, s ∈ 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). Then (E, V) induces a projection p_{⋆+} ∈ L(K) « on ascending edges ».

Problems ^{▶} Θ^{2} ≠ 1 ▶ p_{+⋆} := 1−Θp_{⋆+} Θ^{∗} ≠ p_{⋆+} ^{▶} p_{++} = p_{⋆+} p_{+⋆} ▶ [p_{⋆+} , u_{ij}] 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.

Negative Type Function on A_{o}(Q)

▶ dense sub-*-Hopf algebra S ⊂ S ▶ co-unity ε : S → ℂ, antipode κ : S → S Ex. : S = ℂΓ ⊂ C^{∗}(Γ), ε(g) = 1, κ(g) = g^{−1}

Proposition ▶ π : S → L(H) *-representation. ▶ real π-cocycle : a linear c : S → H st : ▶ c(1) = 0, c(xy) = π(x)c(y) + ε(y)c(x) and ▶ (c(κ(x^{∗}))|c(κ(x)^{∗})) ∈ ℝ. Put φ = ( · | ·) ∘ (c⊗c) ∘ (∗κ⊗id) ∘ δ on S. Then φ(x^{∗}x) ≤ 0 for all x ∈ Ker ε.

Theorem (A_{o}(Q), QQ̄ ∈ ℂid, Tr Q^{∗}Q > 2) Then Ker (T −S) = K_{g}^{⊥} and (T −S)(K_{g}) contains c_{0}(x) = (λ(x) − ε(x)) Λ_{H}(1) for all x ∈ S.

▶ Path and distance to the origin in the quantum Cayley graph