Gamma-Elements for Free Quantum Groups

Gamma-Elements for Free Quantum Groups

Roland Vergnioux

[email protected] 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

I category C with ⊕, ⊗ , ...

I Irr C : irreducible corepresentations

hilbertian objects [Baaj-Skandalis]

– H = L ² ( S, h ), λ : S → L ( H ), S _red = λ ( S ) – decomposition H = ^L _{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 D ¯ = D, 1 _C ∈ D /

– p ₁ = ^P _r∈D p _r

classical graph associated to ( C , D )

– v _{= Irr C ,} e _{= { ( r, r} ⁰ _{) ∈} v ² _{| ∃ s ∈ D r} ⁰ _{⊂ r ⊗ s}}

– the reversing map θ is well defined : r ⁰ ⊂ r ⊗ s ⇐⇒ r ⊂ r ⁰ ⊗ ¯ s

– geometrical edges, orientation

quantum graph associated to ( C, D ) – ` ² -space of vertices : H

– ` 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 ⊗ )V and T = ( ⊗ id)V

nb : Θ ² 6 = id

Ascending Edges

hypothesis

The classical graph of (C, D) is a strict tree, choose the origin 1 _C ^I ascending orientation

proposition In this case the discrete quan- tum group associated to C is a free product of A _o (Q _i )’s and A _u (R _j )’s (with Q _i Q ¯ _i ∈ C _id).

definition (quantum ascending orientation) – p

= ^P {V ^∗ ( p _r ⊗ p _r

) V | ( r, r ⁰ ) ∈ e _{+ }}

– p

:= Θ ^∗ (1 − p

)Θ 6= 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 Q Q ¯ ∈ C _{id and Tr Q} ^∗ _{Q > 2.}

theorem

– T _|K

is injective and its image equals (1 − p ₀ )H .

– p

|K

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

Θ p

)

theorem

p

K _g is closed and its orthogonal is naturally

isomorphic to H _∞ .