Free entropy dimension and the orthogonal free quantum groups

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Free entropy dimension and the orthogonal free quantum groups

Roland Vergnioux

joint work with Michael Brannan

University of Normandy (France)

Trieste, March 1st, 2017

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Introduction

Outline

1 Introduction

Orthogonal free quantum groups The von Neumann algebraL(FO_n)

2 Free entropy dimension Free entropy

The case ofFOn

3 1-boundedness 1-boundedness The case ofFO_n

The quantum Cayley graph

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Introduction Orthogonal free quantum groups

Orthogonal free quantum groups

Wang’s algebradefined by generators and relations:

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

It comes with a natural “group-like” structure:

∆ :A_o(n)→A_o(n)⊗A_o(n),u_ij 7→P

ku_ik⊗u_kj. Why “group-like”?

We have Ao(n)C(On),uij 7→(g 7→gij) and ∆ induces

∆ :C(O_n)→ C(O_n)⊗C(O_n), ∆(f)(g,h) =f(gh).

One can recover the compact group O_n from (C(O_n),∆).

We denote Ao(n) =C(O_n⁺), whereO_n⁺ is a compact quantum group.

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Orthogonal free quantum groups

We have Ao(n)C(On),uij 7→(g 7→gij) and ∆ induces

∆ :C(O_n)→ C(O_n×O_n), ∆(f)(g,h) =f(gh).

One can recover the compact group O_n from (C(O_n),∆).

We denote Ao(n) =C(O_n⁺), whereO_n⁺ is a compact quantum group.

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Orthogonal free quantum groups

We have Ao(n)Cn=C^∗((Z/2Z)^∗n),uij 7→δijbi and ∆ induces

∆ :C_n→C_n⊗C_n,g 7→g⊗g for g ∈C^∗((Z/2Z)^∗n).

One can recover (Z/2Z)^∗n as {u∈U(C_n)|∆(u) =u⊗u}.

We denote Ao(n) =C^∗(FOn), whereFOn is a discrete quantum group.

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Discrete/Compact 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^∗(Γ) =C(G).

General theory : Haar state, Peter-Weyl, Tannaka-Krein...

Theorem (Woronowicz)

There exists a unique state h:C^∗(Γ)→Csuch that (h⊗id)∆ = (id⊗h)∆ = 1⊗h.

Iregular representationλ:C^∗(Γ)→B(H),

Ireduced Woronowicz C^∗-algebra C_red^∗ (Γ) =λ(C^∗(Γ)),

Ivon Neumann algebraL(Γ) =C_red^∗ (Γ)⁰⁰⊂B(H).

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Introduction The von Neumann algebraL(FO_n)

Known results about L ( F O

_n

)

We restrict to the case n≥3.

Free probability:

the elements (√

n u_ij)_i,j≤s are asymptotically free and semi-circular with respect to h asn → ∞[Banica-Collins 2007, Brannan 2014];

free entropy dimension of the generators : δ₀(u) = 1 [Brannan-Collins-V. 2012, 2016].

Von Neumann algebra:

L(FOn) is not injective [Banica 1997]

it is a full and solidII1 factor [Vaes-V. 2007]

it has the HAP and the CBAP [Brannan 2012, Freslon 2013]

it is strongly solid [Isono 2015, Fima-V. 2015]

Question: is L(FO_n) isomorphic to a free group factor?

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Introduction The von Neumann algebraL(FO_n)

Known results about L ( F O

_n

)

We restrict to the case n≥3.

Free probability:

free entropy dimension of the generators : δ0(u) = 1 [Brannan-Collins-V. 2012, 2016].

Question: is L(FO_n) isomorphic to a free group factor?

In L(F_n) with canonical generators a_i one hasδ₀(a) =n.

But it is not known whether δ0 is an invariant of the von Neumann algebra.

However strong1-boundednessis an invariant [Jung 2007]...

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Free entropy dimension

Outline

1 Introduction

The case ofFOn 3 1-boundedness

1-boundedness The case ofFO_n

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Free entropy dimension Free entropy

Free entropy

(M, τ): finite von Neumann algebra with fixed traceτ. H=L²(M, τ).

Fix a tuple of self-adjoint elementsx = (x₁, . . . ,x_m)∈M^m.

χ(x) : microstates free entropy /χ^∗(x) : non microstates free entropy.

Properties of χ andχ^∗: [Voiculescu]

χ(x)∈R∪ {−∞},χ(xi) =RR

ln|s−t|dx_i(s)dxi(t) +C. χ(x)≤χ(x₁) +· · ·+χ(x_m) with equality ifx₁, . . . ,x_m are freely independant.

only forχ: assumingχ(xi)>−∞, equality in the previous point implies free independance.

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Free entropy

Free entropy dimension. [Voiculescu]

Assume M contains a free family s = (s₁, . . . ,s_m) of (0,1)-semicircular elements, also free from x. One defines:

δ0(x) =m−lim infδ→0 χ(x+δs :s)/lnδ δ^∗(x) =m−lim infδ→0 χ^∗(x+δs)/lnδ

δ₀(x) only depends on the algebra generated byx. It is not known whether it only depends on the von Neumann algebra.

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Free entropy

Free entropy dimension. [Voiculescu]

Assume M contains a free family s = (s₁, . . . ,s_m) of (0,1)-semicircular elements, also free from x. One defines:

δ0(x) =m−lim infδ→0 χ(x+δs :s)/lnδ δ^∗(x) =m−lim infδ→0 χ^∗(x+δs)/lnδ

δ₀(x) only depends on the algebra generated byx. It is not known whether it only depends on the von Neumann algebra.

We have the following deep result:

Theorem (Biane-Capitaine-Guionnet 2003) We have χ(x)≤χ^∗(x), henceδ0(x)≤δ^∗(x).

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Free entropy dimension The case ofFO_n

The case of F O

_n

Theorem (Jung 2006)

If W^∗(x) embeds in R^ω and contains a diffuse subalgebra, thenδ₀(x)≥1.

In L(FO_n), the subalgebra (P

u_ii)⁰⁰'L^∞([−2,2]) is diffuse.

[Brannan-Collins-V. 2014]: forn6= 3, L(FO_n) embeds inR^ω. Theorem (Connes-Shlyakhtenko 2005)

We have δ^∗(x)≤β⁽²⁾₁ (A, τ)−β⁽²⁾₀ (A, τ) + 1where A=Chxi.

By diffuseness,β₀⁽²⁾(A,h) =β₀⁽²⁾(FO_n) = 0 [Kyed].

[V. 2012, Kyed-Raum-Vaes-Valvekens 2017]: β₁⁽²⁾(FO_n) = 0.

Corollary

For n>3we haveδ0(u) =δ^∗(u) = 1 inL(FOn).

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1-boundedness

Outline

1 Introduction

The case ofFOn

3 1-boundedness 1-boundedness The case ofFO_n

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1-boundedness 1-boundedness

1-boundedness and von Neumann isomorphisms

Recall that δ^∗(x) =m−lim infδ→0 χ^∗(x+δs)/lnδ.

Hence δ^∗(x)≤α iffχ^∗(x+δs)≤(α−m)|lnδ|+o(lnδ) asδ →0.

One says that x isα-boundedfor δ^∗ if

χ^∗(x+δs)≤(α−m)|lnδ|+K for small δ and some constantK.

There is a similar notion of α-boundedness forδ0 [Jung].

Theorem (Jung 2007)

If x is 1-bounded for δ₀ and χ(x_i)>−∞for at least one i , then any tuple y of generators of W^∗(x)is 1-bounded for δ0 (henceδ0(y)≤1).

In particular if M is generated by a 1-bounded tuple of generators, it is not isomorphic to any free group factor.

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Proving 1-boundedness

Consider the algebra of polynomials inmnon-commuting variablesChXi= ChX₁, . . . ,Xmi. There are unique derivations

δ_i :ChXi →ChXi ⊗ChXi

such thatδi(Xj) =δij(1⊗1), with the bimodule structureP·(Q⊗R)·S = PQ⊗RS. One has e.g.

∂₁(X₂X₁X₃²X₁X₄) =X₂⊗X₃²X₁X₄+X₂X₁X₃²⊗X₄.

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Proving 1-boundedness

Consider the algebra of polynomials inmnon-commuting variablesChXi= ChX₁, . . . ,Xmi. There are unique derivationsδ_isuch thatδ_i(X_j) =δ_ij(1⊗1).

For P = (P₁, . . . ,P_l)∈ChXi^l, put

∂P = (∂jPi)∈ChXi ⊗ChXi ⊗B(C^m,C^l).

Denote H=L²(M, τ). Evaluating atX =x one obtains an operator

∂P(x)∈B(H⊗H⊗C^m,H⊗H⊗C^l),

which commutes to the right action (ζ⊗ξ)·(x⊗y) =ζx⊗yξ ofM⊗M^◦ onH⊗H. One considers the Murray-von Neumann dimension:

rank∂P(x) = dimM⊗M^◦Im∂P(x).

Theorem (Jung 2016, Shlyakhtenko 2016)

Assume that x satisfies the identities P(x) = 0 and that ∂P(x)is of determinant class. Then x is α-bounded for δ0 and δ^∗, with

α=m−rank∂P(x).

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Relations in F O

_n

We take m=n²,X = (X_ij)_ij ∈ChX_iji ⊗M_n(C),x =u = (u_ij)_ij. We consider the l = 2n² canonical relations:

P = (P1,P2) = (X^tX −1,XX^t−1)∈ChXi ⊗Mn(C)^⊕2. Following [Shlyakhtenko 2016] it is easy to prove that:

Proposition

We have n²−rank∂P(u) =β₁⁽²⁾(FOn)−β₀⁽²⁾(FOn) + 1 = 1.

Hence if ∂P(u) is of determinant class, Jung–Shlyakhtenko’s result allows to conclude that u is 1-bounded.

In the case of a discrete group Γ, this would follow from L¨uck’s determinant conjecture, which holds e.g. if Γ is sofic. In the quantum case, there is no such tool (yet?) to prove the determinant conjecture...

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Computation of ∂P (u)

Determinant class: (h⊗h⊗Tr)(ln₊(∂P(u)^∗∂P(u)))>−∞.

Identify M_n(C)'p₁H=Span{u_ijξ₀} ⊂H.

Then u ∈C_red^∗ (FOn)⊗Mn(C) acts by left mult. on H⊗p1H.

IfS :H→H is the antipode, we have inB(H⊗H⊗p1H):

∂P1(u) = (1⊗S⊗S)u23(1⊗S⊗1) +u₁₃^∗

∂P₂(u) = (1⊗S⊗S)u^∗₂₃(1⊗S⊗S) +u₁₃(1⊗1⊗S)

Proposition

We have ∂P1(u)^∗∂P1(u) =∂P2(u)^∗∂P2(u) and it is unitarily conjugated to (2 + 2ReΘ)⊗1∈B(H⊗p₁H⊗H), where

Θ = (S⊗1)u(S⊗S)∈B(H⊗p1H).

Fact: Θ is the reversing operator of the quantum Cayley graph of FO_n!

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1-boundedness The quantum Cayley graph

Decomposing the quantum Cayley graph

Classical case

For Λ = Λ⁻¹⊂Γ, the Cayley graph of (Γ,Λ) is given by X⁽⁰⁾= Γ, X⁽¹⁾= Γ×Λ,

∂:X⁽¹⁾ →X⁽⁰⁾×X⁽⁰⁾,(g,h)7→(g,gh), θ:X⁽¹⁾→X⁽¹⁾,(g,h)7→(gh,h⁻¹).

ConsiderH =`²(Γ),p₁H =`²(Λ),u=diag(λ(g))_g∈Λ,S(g) =g⁻¹. Then:

Θ(g ⊗h) = (S⊗1)u(S⊗S)(g⊗h) =gh⊗h⁻¹. We have Θ² = 1, H⊗p₁H=Ker(Θ−1)⊕Ker(Θ + 1).

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Decomposing the quantum Cayley graph

Classical case

We have Θ² = 1, H⊗p₁H=Ker(Θ−1)⊕Ker(Θ + 1).

Quantum case

We have Θ² 6= 1, Ker(Θ−1)⊕Ker(Θ + 1) H⊗p1H. The description of Ker(Θ±1) was an essential tool in the proof ofβ₁⁽²⁾(FO_n) = 0.

Theorem

OnKer(Θ−1)^⊥∩Ker(Θ + 1)^⊥,Re(Θ)'L

Re(rα)is an infinite direct sum of real parts of weighted right shifts r_α.

Lemma

For any right shift r with weights in ]0,1],2 + 2Rer is of determinant class with respect to the specific state coming from h⊗Tr.

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Conclusion

Finally one can apply Jung-Shlyakhtenko’s result:

Corollary

The generating matrix u is 1-bounded in L(FOn).

L(FO_n) is not isomorphic to a free group factor.

Next questions...

Is there a group Γ such that L(FO_n)'L(Γ)?

What about L(FO(Q)) — the typeIII case?

What about L(FU_n)? Recall thatL(FU₂)'L(F₂).