Stabilizer subgroups of universal compact quantum groups and the Connes embedding property

Stabilizer subgroups of universal compact quantum groups and the Connes embedding property

Roland Vergnioux

joint work with Benoˆıt Collins and Michael Brannan

University of Normandy (France)

Greifswald, march 11th, 2015

Introduction

Outline

1 Introduction

The universal NC orthogonal random matrix Compact and discrete quantum groups Main results

2 Stabilizer subgroups Generating subgroups Stabilizer subgroups of O_n⁺ Idea of the proof

3 Applications

Connes’ embedding property

Free entropy dimension and microstates

Introduction The universal NC orthogonal random matrix

A NC probability space

Underlying algebra defined by generators and relations [Wang]:

Ao(n) =hu_ij,1≤i,j ≤n |uij =u^∗_ij, (uij) unitaryi.

We have a coproduct which allows to convolve states:

∆ :A_o(n)→A_o(n)⊗A_o(n), u_ij 7→P

ku_ik⊗u_kj.

In fact (A_o(n),∆) is a WoronowiczC^∗-algebra Iunique bi-invariant state h :Ao(n)→C, (h⊗id)∆ = (id⊗h)∆ = 1h.

u ∈Mn(C)⊗A_o(n) universaln×n orthogonal matrix with NC entries Haar distributed NC random orthogonal matrix

Ijoint moments of the entriesuij? n→ ∞ asymptotics?

Introduction The universal NC orthogonal random matrix

Old and new results

Some known results about A_o(n) in NC probability:

χ₁ =P

u_ii is a semicircular variable with respect toh [Banica 1997];

the elements (√

n uij)_i,j≤N are asymptotically free and semi-circular with respect to h asn → ∞[Banica-Collins 2007];

computation of the spectral measure ofu_ij with respect to h for n fixed [Banica-Collins-Zinn-Justin 2009];

convergence of (√

n u_ij)_i,j≤N stronglyin distribution [Brannan 2014].

Main result [Brannan-Collins-V.]:

The generators uij admit matricial microstates (if n6= 3).

One consequence:

The (modified, microstate) free entropy dimension δ₀(u_ij) equals 1.

Introduction Compact and discrete quantum groups

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.

Notation : A=C^∗( ) =C(G).

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∆ = ∆.

Introduction Compact and discrete quantum groups

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.

Notation : A=C^∗( ) =C(G).

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 ,

trivial representation / co-unit:C_f^∗( ) =Cf(G)→C, f.-d. corepresentationsv ∈Mk(C)⊗C(G), intertwiners T ∈Hom_G(v,w)⊂M_l,k(C).

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

Introduction Compact and discrete quantum groups

Back to the algebra A

(n)

Recall Wang’s algebra:

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

Consider the discrete group FOn= (Z/2Z)^∗n and the compact groupOn. We have two interesting quotient maps:

Ao(n)→Ao(n)/I 'C^∗(FOn) with I =hu_ij,i 6=ji, A_o(n)→A_o(n)/J 'C(O_n) with J=h[u_ij,u_kl]i.

We denote Ao(n) =C^∗(FOn) =C(O_n⁺) with dual quantum groups:

FO_n, the (discrete) “orthogonal free quantum group”;

O_n⁺, the (compact) “universal orthogonal quantum group”.

Introduction Main results

Analogies with free group C

-algebras

FO_n shares many properties with usual free groups.

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

L(FOn) is a full and strongly solid factor [Vaes-V. 2005, Isono 2012];

Rapid Decay [V. 2007], K-amenability [Voigt 2011], ...

As far as cocycles are concerned:

the “path cocycle” onFO_n is trivial andH₍₂₎¹ (FO_n) = 0 [V. 2012];

Haagerup’s Property [Brannan 2012]: existence of a proper cocyle;

classif. of central generating functionals [Franz-Kula-Cipriani 2014];

there is a proper cocyle living in the adjoint representation, which is weakly contained inλ[Fima-V. 2014].

Introduction Main results

Analogies with free group C

-algebras

FO_n shares many properties with usual free groups.

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

L(FOn) is a full and strongly solid factor [Vaes-V. 2005, Isono 2012];

Rapid Decay [V. 2007], K-amenability [Voigt 2011], ...

Main result of this talk, restated:

L(FO_n) embeds in R^ω (Connes’ embedding property,n≥3).

Strategy:

FO2 is amenable, hence L(FO2)⊂R^ω Iinduction overn.

O_n⁺ is generated by two copies of O_n−1⁺ .

Stabilizer subgroups

Outline

1 Introduction

The universal NC orthogonal random matrix Compact and discrete quantum groups Main results

2 Stabilizer subgroups Generating subgroups Stabilizer subgroups of O_n⁺ Idea of the proof

3 Applications

Connes’ embedding property

Free entropy dimension and microstates

Stabilizer subgroups Generating subgroups

Generating subgroups

G compact quantum group withfull Woronowicz C^∗-algebra C_f(G).

Closed subgroup H⊂G: compact quantum group with surjective Hopf-∗-homomorphismπ :C_f(G)C_f(H).

Inner faithful ∗-homomorphismf :C(G)→B: for any factorization f :C_f(G)−→^π C_f(H)−→^g B

with π surjective Hopf-∗-homomorphism, π is an isomorphism.

Definition

Let (H1, π1), (H2, π2) be closed subgroups of G. Then G=hH1,H2i if (π1⊗π₂)◦∆ :C_f(G)→C_f(H1)⊗C_f(H2) is inner faithful.

More generally, the subgroup H⊂Ggenerated byH1,H2 is theHopf image of (π1⊗π₂)◦∆.

Stabilizer subgroups Generating subgroups

Generating subgroups

Inner faithful ∗-homomorphismf :C(G)→B: for any factorization f :C_f(G)−→^π C_f(H)−→^g B

with π surjective Hopf-∗-homomorphism, π is an isomorphism.

Definition

Let (H1, π₁), (H2, π₂) be closed subgroups of G. Then G=hH1,H2i if (π₁⊗π₂)◦∆ :C_f(G)→C_f(H1)⊗C_f(H2) is inner faithful.

Restriction: v ∈M_n(C)⊗C_f(G) repr. ofG + (H, π) closed subgroup

I(id⊗π)(v), restricted representation of H. Proposition

G=hH1,H2i ⇐⇒ ∀v,w ∈Rep(G)

Hom_G(v,w) =Hom_H1(v,w)∩Hom_H2(v,w)

Stabilizer subgroups Stabilizer subgroups ofO⁺_n

Examples

H₁,H₂⊂G classical compact groups Iusual notions.

G= Γˆ dual of discrete group Γ

Iπi induced by surjective group morphisms πi : Γ→Γi.

IΓˆ =hΓ₁ˆ,Γ₂ˆi ⇐⇒ Γ→Γ₁×Γ₂ faithful.

Some subgroups of O_n⁺:

ρ:C(O_n⁺)→C(O_n), [u_ij,u_kl]→0.

π_i :C(O_n⁺)→C(O_n−1,i⁺ )'C(O_n−1⁺ ),u_ii →1.

Note thatO_n−1,i⁺ ⊂O_n⁺ is the stabilizer ofe_i ∈Cⁿ.

More generally if Gn(C) are the easy quantum groupsassociated to a category of partitions C stable under block removal, we have

Gn−1(C)⊂G_n(C) as stabilizer subgroups.

Stabilizer subgroups Stabilizer subgroups ofO⁺_n

Examples

Some subgroups of O_n⁺:

ρ:C(O_n⁺)→C(O_n), [u_ij,u_kl]→0.

π_i :C(O_n⁺)→C(O_n−1,i⁺ )'C(O_n−1⁺ ),u_ii →1.

Note thatO_n−1,i⁺ ⊂O_n⁺ is the stabilizer ofei ∈Cⁿ.

More generally if Gn(C) are the easy quantum groupsassociated to a category of partitions C stable under block removal, we have

Gn−1(C)⊂Gn(C) as stabilizer subgroups.

Theorem

For n≥4and i 6=j we have O_n⁺ =hO_n−1,i⁺ ,O_n−1,j⁺ i=hO_n−1,i⁺ ,Oni.

Stabilizer subgroups Idea of the proof

Brauer diagrams

P2(k,l): set of partitions of k upper points andl lower points into pairs NC₂(k,l)⊂P₂(k,l): pair partitions that can be represented by a planar diagram with noncrossing strings

Let H=Cⁿ and associate to p ∈P(k,l) the linear map Tp:H^⊗k →H^⊗l:

Tp(ei1⊗ · · · ⊗e_ik) =X

j

i1. . .ik

p j1. . .jl

!

ej1⊗ · · · ⊗e_jl,

where the middle symbol is 1 if all blocs in p join pairs of equal indices, and 0 if not.

Then:

Hom_On(u^⊗k,u^⊗l) =Span{T_p|p ∈P₂(k,l)} [Brauer], Hom_O⁺

n(u^⊗k,u^⊗l) =Span{T_p|p ∈NC2(k,l)} [Banica].

Stabilizer subgroups Idea of the proof

A lemma of linear algebra

Denote TC₂(k,l)⊂P₂(k,l) the subset of diagrams where crossings are allowed only with lines that are connected to an upper point. Then:

Lemma Hom_O⁺

n−1,i(1,u^⊗k) =Span{T_p(e_i⊗ · · · ⊗e_i)|s ≤k,p∈TC₂(s,k)}

Putξ_s =e₁⊗ · · · ⊗e₁⊗e₂+e₁⊗ · · · ⊗e₂⊗e₁+· · ·+e₂⊗e₁⊗ · · · ⊗e₁ ∈H^⊗s. Lemma

We have Hom_O⁺

n−1,i(1,u^⊗k)∩Hom_O⁺

n−1,j(1,u^⊗k) =Hom_O⁺

n(1,u^⊗k) iff the family of vectors {T_p(ξ_s)|1≤s ≤k,p ∈TC₂(s,k)}is linearly independant.

Stabilizer subgroups Idea of the proof

A lemma of linear algebra

Putξ_s =e₁⊗ · · · ⊗e₁⊗e₂+e₁⊗ · · · ⊗e₂⊗e₁+· · ·+e₂⊗e₁⊗ · · · ⊗e₁ ∈H^⊗s. Lemma

We have Hom_O⁺

n−1,i(1,u^⊗k)∩Hom_O⁺

n−1,j(1,u^⊗k) =Hom_O⁺

n(1,u^⊗k) iff the family of vectors {T_p(ξs)|1≤s ≤k,p ∈TC2(s,k)}is linearly independant.

Lemma

If n ≥4, the linear independance property of the previous lemma is true for any k. As a result O_n⁺=hO_n−1,i⁺ ,O_n−1,j⁺ i.

Moreover we have strong numerical evidence of:

Conjecture

The same is true for n= 3.

Applications

Outline

1 Introduction

The universal NC orthogonal random matrix Compact and discrete quantum groups Main results

2 Stabilizer subgroups Generating subgroups Stabilizer subgroups of O_n⁺ Idea of the proof

3 Applications

Connes’ embedding property

Free entropy dimension and microstates

Applications Connes’ embedding property

Connes’ embedding property

Let R^ω be an ultrapower of the hyperfiniteII₁ factor.

For AunitalC^∗-algebra, define

CEP(A) ={τ :A→Ctracial state|πτ(A)⁰⁰,→R^ω tracially}, where π_τ is the GNS representation.

For unimodular discrete quantum group: CEP( ) =CEP(C_f^∗( )).

We say that ishyperlinear ifh∈CEP( ), i.e. if its von Neumann algebra L( ) embeds tracially in R^ω.

Proposition

Ifτ₁,τ₂ ∈CEP( )thenτ₁∗τ₂ = (τ₁⊗τ₂)◦∆∈CEP( ).

Ifτ_n→τ pointwise andτ_n∈CEP( )thenτ ∈CEP( ).

Applications Connes’ embedding property

Connes’ embedding property

We say that ishyperlinear ifh∈CEP( ), i.e. if its von Neumann algebra L( ) embeds tracially in R^ω.

Proposition

Ifτ1,τ2 ∈CEP( )thenτ1∗τ2 = (τ1⊗τ₂)◦∆∈CEP( ).

Ifτn→τ pointwise andτn∈CEP( )thenτ ∈CEP( ).

Let (H1, π₁), (H2, π₂) be subgroups of G.

Denote hi =h_Hi ◦πi :Cf(G)→Candh=h_G :Cf(G)→C.

Proposition

We have G=hH1,H2iiff h = lim(h1∗h2)^∗n pointwise.

Corollary

IfG=hH1,H2i andHˆ1,Hˆ2 are hyperlinear, then Gˆ is hyperlinear.

Applications Connes’ embedding property

Hyperlinearity of F O

Corollary

IfG=hH1,H2i andHˆ1,Hˆ2 are hyperlinear, then Gˆ is hyperlinear.

Recall that FO_n= ˆO_n⁺ andO_n⁺ =hO_n−1,i⁺ ,O_n−1,j⁺ i forn ≥4.

Moreover FO2 is hyperlinear because it is amenable.

IFO_n hyperlinear for alln ifO₃⁺=hO_2,i⁺,O_2,j+i.

Bypass to avoid the use of the conjecture at n= 3:

Lemma (after A. Chirvasitu) We have O₄⁺=hO₂⁺ˆ∗O₂⁺,O4i.

Altogether:

Theorem

FOn is hyperlinear for all n6= 3.

Applications Free entropy dimension and microstates

Free entropy dimension

Denote by δ₀ Voiculescu’s modified free entropy dimension.

Consequence of Connes’ embedding property: we can apply Jung’s

“hyperfinite monotonicity” result. Since L(FOn) contains diffuse von Neumann subalgebras this yields:

Corollary

For the generators uij ofL(FOn), n6= 3, we have1≤δ0(uij).

On the other hand we have an upper bound coming from `²-Betti numbers. More precisely

δ₀(u_ij)≤δ^∗(u_ij)≤β₁⁽²⁾(FO_n)−β₀⁽²⁾(FO_n) + 1 by [Biane-Capitaine-Guionnet] and [Connes-Shlyakhtenko].

Applications Free entropy dimension and microstates

Free entropy dimension

Denote by δ₀ Voiculescu’s modified free entropy dimension.

Consequence of Connes’ embedding property: we can apply Jung’s

“hyperfinite monotonicity” result. Since L(FOn) contains diffuse von Neumann subalgebras this yields:

Corollary

For the generators uij ofL(FOn), n6= 3, we have1≤δ0(uij).

On the other hand we have an upper bound coming from `²-Betti numbers. More precisely

δ₀(u_ij)≤δ^∗(u_ij)≤β₁⁽²⁾(FO_n)−β₀⁽²⁾(FO_n) + 1 by [Biane-Capitaine-Guionnet] and [Connes-Shlyakhtenko].

Applications Free entropy dimension and microstates

Free entropy dimension

On the other hand we have an upper bound coming from `²-Betti numbers. More precisely

δ0(uij)≤δ^∗(uij)≤β₁⁽²⁾(FOn)−β₀⁽²⁾(FOn) + 1

by [Biane-Capitaine-Guionnet] and [Connes-Shlyakhtenko]. Moreover Theorem (V. 2012)

We have β₁⁽²⁾(FO_n) = 0 for all n≥3.

Since FO_n is infinite we have β₀⁽²⁾(FO_n) = 0 [Kyed] and finally Corollary

For the generators uij ofL(FOn), n6= 3, we haveδ0(uij) = 1.

Applications Free entropy dimension and microstates

Microstates

Connes’ embedding property is equivalent to the existence of matricial microstates for the generators uij. More precisely, for every p∈Nand >0 there existsk ∈N and matricesa_ij ∈M_k(C)sa such that

|tr(a_i1j1· · ·aiqjq)−h(ui1j1· · ·uiqjq)| ≤ for all i,j ∈ {1, . . . ,n}^q,q ≤p.

Using the stabilizer subgroups one can write down an explicit construction:

explicit microstate

for O₂⁺=SU−1(2) =⇒ explicit microstate for{u_ij} ⊂O_n⁺

Questions : construct “natural” microstates / a “natural” asymptotic random matrix model for O_n⁺ ?