Gamma-Elements for Free Quantum Groups

Roland Vergnioux vergniou@math.jussieu.fr

August 18, 2003

Motivations

– K-amenability [Cuntz]

– groups acting on trees [Julg-Valette]

– amalgamated free products of amenable discrete quantum groups Compact Quantum Groups [Woronowicz]

main examples

– discrete groups

– free quantum groups [Wang-Banica]

C^{∗}-algebras

– S unital C^{∗}-algebra

– coproduct δ : S → S ⊗S There exists a Haar state h ∈ S^{∗}.

corepresentations v ∈ L(H_{v})⊗S st (id⊗δ)(v) = v_{12} v_{13} ▶ category C with ⊕, ⊗, ... ▶ Irr C : irreducible corepresentations

hilbertian objects [Baaj-Skandalis]

– H = L^{2}(S, h), λ : S → L(H), S_{red} = λ(S)

– decomposition H = ⨁_{r∈Irr C} p_{r}H

– Kac system (H, V, U)

Quantum Caley Graphs

data

– quant. discrete group : S, C, (H, V, U)

– D ⊂ Irr C finite st D̄ = D, 1_{C} ∉ D

– p_{1} = ∑_{r∈D} p_{r}

classical graph associated to (C, D)

– 𝔳 = Irr C, 𝔢 = {(r, r′) ∈ 𝔳^{2} | ∃ s ∈ D r′ ⊂ r⊗s} – the reversing map θ is well defined :

r′ ⊂ r⊗s ⟺ r ⊂ r′⊗s̄ – geometrical edges, orientation

quantum graph associated to (C, D)

– ℓ^{2}-space of vertices : H

– ℓ^{2}-space of edges : K = H ⊗p_{1}H

– Θ = Σ(1⊗U)V (U ⊗U)Σ, K_{g} = Ker (Θ − id)

– V : K → H ⊗H « endpoints » operator

– S = (id⊗ϵ)V and T = (ϵ⊗id)V nb : Θ^{2} ≠ id

Ascending Edges

hypothesis The classical graph of (C, D) is a strict tree, choose the origin 1_{C} ^{▶} ascending orientation

proposition In this case the discrete quantum group associated to C is a free product of A_{o}(Q_{i})’s and A_{u}(R_{j})’s (with Q_{i}Q̄_{i} ∈ ℂid).

definition (quantum ascending orientation)

– p_{⋆+} = ∑ {V^{∗}(p_{r} ⊗p_{r′})V | (r, r′) ∈ 𝔢_{+} }

– p_{+⋆} := Θ^{∗}(1 − p_{⋆+} )Θ ≠ p_{⋆+} !

– p_{⋆−} = 1 − p_{⋆+} , p_{−⋆} = 1 − p_{+⋆}

– p_{++} = p_{+⋆} p_{⋆+} , p_{+−} = ⋯, K_{++} = p_{++} K

definition (quantum Julg-Valette operator)

F_{g}^{∗} : K_{g} ^{p_{++}} K_{++} ^{T} H

Space of Edges at Infinity

We consider the quantum Cayley tree of A_{o}(Q), with QQ̄ ∈ ℂid and Tr Q^{∗}Q > 2.

theorem

– T_{|K_{++}} is injective and its image equals (1 − p_{0})H.

– p_{++} _{|K_{g}} is injective and its image equals {ζ ∈ K_{++} | p_{+−} Θp_{++} ζ ∈ Im (id − p_{+−} Θp_{+−} )}.

definition p_{+−} Θp_{+−} is a « right shift », we put

H_{∞} = lim_{⟶} ((p_{k} ⊗id)K_{+−} , p_{+−} Θp_{+−} )

theorem p_{++} K_{g} is closed and its orthogonal is naturally isomorphic to H_{∞}.