On the adjoint representation of orthogonal free quantum groups

On the adjoint representation of orthogonal free quantum groups

Roland Vergnioux joint work with Pierre Fima

University of Normandy (France) University Paris 7 (France)

Indianapolis, sept. 28th, 2013

Introduction

Outline

Introduction Orthogonal free quantum groups

Orthogonal free quantum groups

Consider the unitalC^∗-algebras defined by generators and relations:

Co(n) =hu_i,1≤i ≤n |ui =u_i^∗, ui unitaryi, A_o(n) =hu_ij,1≤i,j ≤n |u_ij =u^∗_ij, (u_ij) unitaryi.

We recognize C_o(n) =C^∗(FO_n) whereFO_n= (Z/2Z)^∗n.

We denote A_o(n) =C^∗(FO_n). A_o(n) was introduced by S. Wang.

The full structure of FOn is reflected by a coproduct

∆ :C_o(n)→C_o(n)⊗C_o(n),u_i 7→u_i⊗u_i. Similarly there is a natural coproduct

∆ :Ao(n)→Ao(n)⊗A_o(n),uij 7→P

kuik⊗u_kj.

IFO_n is a discrete quantum group : the ortogonal free quantum group.

It is the Pontrjagin dual of the compact quantum group O_n⁺.

Introduction Discrete quantum groups

Discrete quantum groups

A WoronowiczC^∗-algebra is a unitalC^∗-algebraA with ∗-homomorphism

∆ :A→A⊗A(coproduct) such that (∆⊗id)∆ = (id⊗∆)∆,

∆(A)(1⊗A) and ∆(A)(A⊗1) are dense in A⊗A.

Examples :

G compact group, A=C(G), ∆(f) = ((x,y)7→f(xy)), characterized by commutativity of A;

Γ discrete group, A=C^∗(Γ), ∆(g) =g⊗g — but also A=C_red^∗ (Γ), characterized by co-commutativity : Σ∆ = ∆.

Notation : A=C^∗( ).

Introduction Discrete quantum groups

Discrete quantum groups

A WoronowiczC^∗-algebra is a unitalC^∗-algebraA with ∗-homomorphism

∆ :A→A⊗A(coproduct) such that (∆⊗id)∆ = (id⊗∆)∆,

∆(A)(1⊗A) and ∆(A)(A⊗1) are dense in A⊗A.

General theory :

Haar state h∈C^∗( )^∗ IGNS representationλ:C^∗( )→B(`² ), C_red^∗ ( ) =λ(C^∗( )) is again a Woronowicz C^∗-algebra,

L( ) =C_red^∗ ( )⁰⁰ von Neumann algebra of , right regular representation ρ:C^∗( )→B(`² ),

adjoint representationad= (λ, ρ)◦∆ :C_full^∗ ( )→B(`² ), trivial representation :C_full^∗ ( )→C,

is called unimodular if h is a trace, amenable if factors through λ.

Introduction Main results

Analogies with free group C

-algebras

FO_n shares many (analytical) properties with usual free groups:

FOn is non amenable for n≥3 [Banica 1997];

C_red^∗ (FOn) is simple,L(FOn) is a full factor [Vaes-V. 2005];

FO_n is K-amenable [Voigt 2009];

FO_n is a-T-menable [Brannan 2011];

later in this talk : rapid decay, weak amenability, bi-exactness, ...

However : the first `²-Betti number of FOn vanishes [V. 2009].

Question : where do proper cocycles live ? Main result of this talk [Fima-V.] :

ad ≺λfor FO_n,

deformation of the identity by automorphisms.

Applications : fullness, property (HH), strong solidity...

The adjoint representation

Outline

The adjoint representation Some classical results

Classical results

Fn : free group onn generators. `(g) : length of g ∈Fn. Recall the main results of [Haagerup 1979] :

rapid decay : for x∈C^∗(Fn) supported on elements of lengthk, kλ(x)k ≤(k+ 1)kxk₂, where kxk²₂ =h(x^∗x).

a-T-menability : (g 7→e^−t`(g)g) defines a completely positive map T_t :C_red^∗ (F_n)→C_red^∗ (F_n) for all t>0.

Corollaries :

metric approximation property (MAP) for C_red^∗ (Fn) : there exists M_α :C_red^∗ (F_n)→C_red^∗ (F_n) contractive with finite rank such that M_α(x)→x for all x.

states ϕ∈C^∗(F_n)^∗₊ factor throughλiff (g 7→ϕ(g)e<sup>−t`(g)</sup>) is in `²(F_n) for all t>0.

The adjoint representation Some classical results

Classical results

Application to ad:C^∗(F_n)→B(`²F_n).

The vectorξ0=δe ∈`²(Fn) is fixed Iad^◦=ad onξ₀^⊥. Considerϕ:x7→(δg|ad(x)δg) onC^∗(Fn).

We have ϕ(h) = 1 if hg =gh,ϕ(h) = 0 else.

But C(g) ={h∈F_n|hg =gh}is cyclic forg 6=e:

C(g) ={w^k |k ∈Z} with w =uvu⁻¹,v cyclically reduced.

Inon-zero values ofϕ(h)e^−t`(h) : e^−t(|k|p+q), forh =w^k. Haagerup’s characterization Iϕ≺λfor g 6=e.

Conclusion : ad^◦ ≺λ.

The adjoint representation Factorization throughC_red^∗ (FOn)

The quantum case

There is a natural “word length” on FO_n : `²FO_n=L

p_k`²FO_n. Definition : L

l≤kpl`²FOn=Span{P(uij)ξ0 |degP ≤k}.

Property of Rapid Decay :

Theorem (V. 2004)

If x ∈C^∗(FO_n) is such thatλ(x)ξ₀ ⊂p_k`²FO_n, then kλ(x)k ≤(2k+ 5)kxk₂.

The adjoint representation Factorization throughC_red^∗ (FOn)

The quantum case

There is a natural “word length” on FO_n : `²FO_n=L

p_k`²FO_n. Definition : L

l≤kpl`²FOn=Span{P(uij)ξ0 |degP ≤k}.

A-T-menability :

Denote U_k the Chebyshev polynomials of the second kind.

Theorem (Brannan 2011)

For all t ∈]2,n], the formula Tt(x)ξ0 =P

k Uk(t/2) Uk(n/2) pkxξ0

defines a completely positive map T_t :C_red^∗ (FO_n)→C_red^∗ (FO_n).

The adjoint representation Factorization throughC_red^∗ (FOn)

The quantum case

There is a natural “word length” on FO_n : `²FO_n=L

p_k`²FO_n. Definition : L

l≤kpl`²FOn=Span{P(uij)ξ0 |degP ≤k}.

A-T-menability :

Denote U_k the Chebyshev polynomials of the second kind.

Theorem (Brannan 2011)

For all t ∈]2,n], the formula Tt(x)ξ0 =P

k Uk(t/2) Uk(n/2) pkxξ0

defines a completely positive map T_t :C_red^∗ (FO_n)→C_red^∗ (FO_n).

Define`∈C^∗(FOn)^∗ (unbounded) by:

`(x) =k(x) ifλ(x)ξ<sub>0</sub> ∈p<sub>k</sub>`²(FO_n).

Corollary : statesϕ∈C^∗(FO_n)^∗₊ factor through λiffϕe^−t` is continuous with respect to k · k₂ for allt >0.

The adjoint representation The adjoint representation ofFOn

On the adjoint representation

The line Cξ0⊂`² is invariantiff is unimodular.

Iwe can still considerad^◦=ad onξ₀^⊥⊂`²(FO_n).

Theorem (Fima-V. 2012) We have ad^◦ ≺λfor FOn.

Proof : use the preceeding criterium.

However : no combinatorial property of centralizers as in the classical case.

Instead : growth estimates for ϕ:x 7→(ξ|ad(x)ξ), ξ∈pk`²(FOn),k ≥1, using computations in the category of corepresentations of FOn.

The adjoint representation The adjoint representation ofFOn

On the adjoint representation

The line Cξ0⊂`² is invariantiff is unimodular.

Iwe can still considerad^◦=ad onξ₀^⊥⊂`²(FO_n).

Theorem (Fima-V. 2012) We have ad^◦ ≺λfor FOn.

Proof : use the preceeding criterium.

However : no combinatorial property of centralizers as in the classical case.

Instead : growth estimates for ϕ:x 7→(ξ|ad(x)ξ), ξ∈pk`²(FOn),k ≥1, using computations in the category of corepresentations of FOn.

First application :

Corollary (Vaes-V. 2005)

For n≥3, the representationad^◦ has spectral gap : 6≺ad^◦. In particular FOn is not inner amenable and L(FOn) is a full factor.

A Deformation by automorphisms

Outline

A Deformation by automorphisms The deformation

The deformation

Action of On

By definition there is a surjective map π :C^∗(FO_n) =C(O_n⁺)→C(O_n).

By Fell’s absorption principle, ∆ factors to

∆⁰:C_red^∗ (FOn)→C^∗(FOn)⊗C_red^∗ (FOn).

We obtain an action of On onC_red^∗ (FOn) by automorphisms : α_g = ((ev_g ◦π)⊗id)◦∆ :C_red^∗ (FO_n)→C_red^∗ (FO_n).

Deformation of C_red^∗ (FOn) inside a bigger algebra Put C =C_red^∗ (FO_n) and ˜C =C_red^∗ (FO_n)⊗C_red^∗ (FO_n)

ι= ∆_red:C →C˜ the natural embedding E : ˜C →ι(C) unique trace-pres. cond. exp.

We deform ιby putting A_g(x) = (id⊗α_g)ι:C →C˜ forg ∈O_n.

A Deformation by automorphisms The deformation

The deformation

Deformation of C_red^∗ (FO_n) inside a bigger algebra Put C =C_red^∗ (FOn) and ˜C =C_red^∗ (FOn)⊗C_red^∗ (FOn)

ι= ∆_red:C →C˜ the natural embedding E : ˜C →ι(C) unique trace-pres. cond. exp.

We deform ιby putting Ag(x) = (id⊗α_g)ι:C →C˜ forg ∈On.

Proposition (Fima-V. 2012)

We have E ◦A_g =T_t :C_red^∗ (FO_n)→C_red^∗ (FO_n), where t =Tr(g).

Irecover complete positivity of Brannan’s deformation.

Iget deformation ofC ⊂C˜ by 1-param. group of autom. (Ags)s∈R.

A Deformation by automorphisms Application to cocycles

Application to cocycles

Recall : a-T-menability ⇔existence of a proper cocycle in some repr. π.

Classical case F_n : natural proper cocyclec given by paths in the Cayley graph. In that case π=L

2nλ.

[Brannan 2011] Iproper cocycle for FOn. What can be said about π ? Theorem (V. 2009)

For n≥3we have H¹(C[FO_n], `²(FO_n)) = 0.

All cocycles in (finite sums of) λare trivial.

A Deformation by automorphisms Application to cocycles

Application to cocycles

[Brannan 2011] Iproper cocycle for FOn. What can be said about π ? Concrete construction of a proper cocycle forFOn

Differentiate the deformationA_g : get for all X ∈o_n

Ia derivation δ_X :C[FO_n]→C˜ ι(C)

Ia cocycle cX :C[FOn]→`²(FOn) Cξ0

Proposition (Fima-V. 2013)

For all X ∈o_n, X 6= 0, c_X is proper. The representation of FO_n corresponding to the bimodule _ι(C)C˜_ι(C) is ad.

In particular the cocycle arising from Brannan’s deformation can be realized inside π=ad^◦.

Corollary (Fima-V. 2013)

FO_n satisfies Property strong (HH) form [Ozawa-Popa 2008].

A Deformation by automorphisms Application to strong solidity

Application to strong solidity

Recall M is stronly solid if for every diffuse amenable P ⊂M, the normalizer NM(P) generates an amenable vN subalgebra.

strongly solid + non-amenable⇒prime + no Cartan subalgebra . [Chifan-Sinclair 2011, Popa-Vaes 2012] CBAP + AO⁺ ⇒strongly solid Theorem (V. 2004)

FOn satisifies a strong Akemann-Ostrand Property (AO⁺).

Theorem (Freslon 2012)

FOn is weakly amenable (CBAP) with constant 1.

Theorem (Isono 2012) L(FO_n) is strongly solid.

A Deformation by automorphisms Application to strong solidity

Application to strong solidity

Recall M is stronly solid if for every diffuse amenable P ⊂M, the normalizer NM(P) generates an amenable vN subalgebra.

strongly solid + non-amenable⇒prime + no Cartan subalgebra . [Ozawa-Popa 2008] CBAP + strong (HH) ⇒ strongly solid

Corollary (Fima-V. 2013) L(FO_n) is strongly solid.