Furstenberg boundary for discrete quantum groups

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Furstenberg boundary for discrete quantum groups

Roland Vergnioux

joint work with M. Kalantar, P. Kasprzak, A. Skalski

University of Normandy (France)

Seoul, April 7, 2021

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Introduction

Outline

1 Introduction Motivation

Discrete quantum groups Actions

Orthogonal free quantum groups

2 Boundary actions Γ-boundaries

Boundaries and unique stationarity The Gromov boundary ofFOQ

An FO_Q-boundary

3 Applications

Uniqueness of trace

Universal boundary and the amenable radical

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Introduction Motivation

Motivation

Inotion of Γ-boundaryin topological dynamics (Furstenberg, 1950s)

Isurprising connection with the structure ofreduced group C^∗-algebra (Kalantar-Kennedy, Breuillard-Kalantar-Kennedy-Ozawa, 2010s) Γ discrete group Itranslation operatorsλ(g)∈B(`²Γ)

IreducedC^∗-algebraC_red^∗ (Γ) =Span {λ(g),g ∈Γ}

Iwith traceh(x) = (11e|x11e),h(λ(g)) =δg,e

Trace : ϕ:C_red^∗ (Γ)→C, positive, unital, ϕ(xy) =ϕ(yx).

Theorem (BKKO)

C_red^∗ (Γ)simple ⇔ ∃freeΓ-boundary ΓyX .

C_red^∗ (Γ)has a unique trace⇔ ∃ faithfulΓ-boundary ΓyX .

In particular simplicity ⇒ uniqueness of trace for reduced C^∗-algebras of discrete groups. The converse is false.

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Introduction Discrete quantum groups

Discrete quantum groups

A discrete quantum group Γis given by : a von Neumann algebra `^∞(Γ) =L`^∞

α∈IB(H_α) with dimH_α <∞ a normal∗-homomorphism ∆ :`^∞(Γ)→`^∞(Γ) ¯⊗`^∞(Γ) such that (∆⊗id)∆ = (id⊗∆)∆ (coproduct)

left and right ∆-invariant nsf weightsh_L,h_R on`^∞(Γ) Γ isunimodular if h_L=h_R. Denote`²(Γ) =L²(`^∞(Γ),h_L).

Classical case : Γ= Γ =I,`^∞(Γ) =`^∞(Γ), ∆(f) = ((r,s)7→f(rs)), hL(f) =hR(f) =P

r∈Γf(r).

In general : coproduct Itensor product π⊗ρ:= (π⊗ρ)∆ for representations π,ρ of `^∞(Γ) Itensor C^∗-categoryCorep(Γ).

I : irreducible objects up to equivalence.

The multiplication table of Γ is replaced by the spaces Hom(α, β⊗γ).

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Introduction Actions

Actions

Canonical dense subalgebra : c0(Γ)⊂`^∞(Γ) given by Lc0

α∈I. A Γ-C^∗-algebra is aC^∗-algebraA equipped with a ∗-homomorphism α:A→M(c₀(Γ)⊗A) such that (id⊗α)α= (∆⊗id)α (coaction).

For a∈A,ν ∈A^∗,µ∈c₀(Γ)^∗ we can then define L_µ(a) = (µ⊗id)α(a)∈M(A), Pν(a) = (id⊗ν)α(a)∈`^∞(Γ),

µ∗ν= (µ⊗ν)α∈A^∗.

A Γ-mapT :A→B is a linear map such thatT ◦L_µ=L_µ◦T. Classical case : ΓyX,A=C₀(X),α(f) = ((r,x)7→f(r·x)).

Example : A=c₀(Γ),α= ∆ “translation action”.

By invariance, the maps Lµ extend to bounded operators on`²(Γ).

IC^∗-algebra C_red^∗ (Γ) =Span {L_µ}with state h= (ξ0 | ·ξ0).

Note : h is a trace⇔ Γunimodular.

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Introduction Actions

Actions

Canonical dense subalgebra : c0(Γ)⊂`^∞(Γ) given by Lc0

α∈I. A Γ-C^∗-algebra is aC^∗-algebraA equipped with a ∗-homomorphism α:A→M(c₀(Γ)⊗A) such that (id⊗α)α= (∆⊗id)α (coaction).

For a∈A,ν ∈A^∗,µ∈c₀(Γ)^∗ we can then define L_µ(a) = (µ⊗id)α(a)∈M(A), Pν(a) = (id⊗ν)α(a)∈`^∞(Γ),

µ∗ν= (µ⊗ν)α∈A^∗.

A Γ-mapT :A→B is a linear map such thatT ◦L_µ=L_µ◦T. Definition

The cokernel Nα ⊂`^∞(Γ) ofα is the weak closure of

{P_ν(a),a∈A, ν ∈A^∗}. We say that α is faithful if N_α=`^∞(Γ).

We have ∆(Nα)⊂Nα⊗N¯ _α. In the classical case this impliesNα=`^∞(Γ)^Λ with ΛCΓ, and we have Λ =Kerα in this case. In the quantum caseN_α is not necessarily associated to a subgroup Λ<Γ...

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Introduction Orthogonal free quantum groups

Orthogonal free quantum groups

Let N∈N,Q∈GL_N(C) s.t. QQ¯ =±I_N.

The discrete quantum groupΓ=FO(Q) can be described as follows:

ICorep(FO(Q)) is the Temperley-Lieb category withδ =Tr(Q^∗Q),

II =N withk⊗1'1⊗k '(k−1)⊕(k+ 1), ¯k =k,

IH₀ =C,H₁=C^N andHom(0,1⊗1) =Ct₁ with t₁ =P

e_i ⊗Qe_i. We can then constructHk by induction,`^∞(FOQ) and compute ∆.

Assume Q =I_N — we writeFO_Q =FO_N.

ω_ij = (e_i | ·e_j)∈B(H₁)^∗ ⊂c₀(Γ)^∗ Ioperators L_ij :=L_ω_ij ∈C_red^∗ (FO_N)

ImatrixL= (Lij)ij ∈MN(C_red^∗ (FON)) s.t. L^∗_ij =Lij and LL^∗ =L^∗L=IN Irepresentation of Wang’s algebra:

A_o(N) =C^∗h1,u_ij |u_ij^∗ =u_ij,uu^∗ =u^∗u =I_ni In fact A_o(N) is the full C^∗-algebra ofFO_N...

The terminology comes from the following “classical” quotients of Ao(N):

A_o(N)/(u_ij,i 6=j)'C^∗(F_N), A_o(N)/([u_ij,u_kl])'C(O_N).

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Boundary actions

Outline

An FO_Q-boundary

3 Applications

Uniqueness of trace

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Boundary actions Γ-boundaries

Γ-boundaries

Classical case: ΓyX compact.

We have X ⊂Prob(X) via Dirac measures and ΓyProb(X).

The action ΓyX is:

minimal if ∀x,y ∈X ∃g_n∈Γ s.t. limgn·x =y, in other words: ∀x ∈X Γ·x=X ;

proximal if∀x,y ∈X ∃g_n∈Γ s.t. limg_n·x = limg_n·y ; strongly proximal if ΓyProb(X) proximal,

or equivalently: ∀ν ∈Prob(X) Γ·ν∩X 6=∅.

X is a Γ-boundaryif it is minimal and strongly proximal, or equivalently: ∀ν∈Prob(X) X ⊂Γ·ν.

Classical examples:

G connected simple Lie group, H <G maximal amenable, X =G/H Γ non elementary hyperbolic, X =∂_GΓ Gromov boundary

i) ∀ν ∈Prob(X) X ⊂Γ·ν ii) ∀ν ∈Prob(X) Prob(X) =

iii) ∀ν ∈Prob(X),f ∈C(X)_sa kfk= sup_{µ∈Prob(Γ)} iv) ∀ν ∈Prob(X) Pν is isometric on C(X)sa

v) ∀ν ∈Prob(X) P_ν is a complete isometry

vi) all UCP Γ-mapsT :C(X)→`^∞(Γ) are complete isometries vii) all UCP Γ-mapsT :C(X)→B are complete isometries

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Γ-boundaries

The following assertions are equivalent:

i) ∀ν ∈Prob(X) X ⊂Γ·ν

ii) ∀ν ∈Prob(X) Prob(X) =ConvΓ·ν

iii) ∀ν ∈Prob(X),f ∈C(X)sa kfk= sup_{µ∈Prob(Γ)} iv) ∀ν ∈Prob(X) P_ν is isometric on C(X)_sa

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Γ-boundaries

ii) ∀ν ∈Prob(X) Prob(X) ={µ∗ν, µ∈Prob(Γ)}

iii) ∀ν ∈Prob(X),f ∈C(X)sa kfk= sup_{µ∈Prob(Γ)} iv) ∀ν ∈Prob(X) P_ν is isometric on C(X)_sa

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Γ-boundaries

iii) ∀ν ∈Prob(X),f ∈C(X)sa kfk= sup_{µ∈Prob(Γ)} |hµ∗ν,fi|

iv) ∀ν ∈Prob(X) P_ν is isometric on C(X)_sa v) ∀ν ∈Prob(X) P_ν is a complete isometry

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Γ-boundaries

iii) ∀ν ∈Prob(X),f ∈C(X)sa kfk= sup_{µ∈Prob(Γ)} |hµ,Pν(f)i|

iv) ∀ν ∈Prob(X) P_ν is isometric on C(X)_sa v) ∀ν ∈Prob(X) P_ν is a complete isometry

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Γ-boundaries

iv) ∀ν ∈Prob(X) P_ν is isometric onC(X)_sa

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Γ-boundaries

iv) ∀ν ∈Prob(X) P_ν is isometric onC(X)_sa v) ∀ν ∈Prob(X) P_ν is a complete isometry

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Γ-boundaries

vi) all UCP Γ-mapsT :C(X)→`^∞(Γ) are complete isometries (indeed T =P_ν forν =◦T)

vii) all UCP Γ-mapsT :C(X)→B are complete isometries

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Γ-boundaries

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Γ-boundaries

vi) all UCP Γ-mapsT :C(X)→`^∞(Γ) are complete isometries vii) all UCP Γ-mapsT :C(X)→B are complete isometries Quantum case: ΓyAunital, Prob(X)/Prob(Γ)IS(A)/S(c₀(Γ)) i) no meaning ; ii)–iv) still equiv. ; only iv)⇒v) ; v)–vii) still equiv.

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Boundary actions Boundaries and unique stationarity

Boundaries and unique stationarity

Definition

A unital Γ-C^∗-algebraA is a Γ-boundary if every UCPΓ-mapT :A→B is automatically UCI.

This has good categorical properties : C,→Ais an “essential extension”

in the category of unital Γ-C^∗-algebras with UCPΓ-maps as morphisms and UCI Γ-maps as embeddings.

Choose µ∈S(c0(Γ)). A stateν ∈S(A) is µ-stationary ifµ∗ν =ν.

Proposition (Kalantar)

Assume that A admits a unique µ-stationary state ν and that P_ν is completely isometric. Then A is a Γ-boundary.

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Boundary actions Boundaries and unique stationarity

Boundaries and unique stationarity

Choose µ∈S(c0(Γ)). A stateν ∈S(A) is µ-stationary ifµ∗ν =ν.

Proposition (Kalantar)

Assume that A admits a unique µ-stationary state ν and that P_ν is completely isometric. Then A is a Γ-boundary.

Proof. ν is stationaryiffP_ν(A)⊂H_µ^∞(Γ) :={f ∈`^∞(Γ)|L_µ(f) =f}.

Then Pν is the unique UCP Γ-mapA→H_µ^∞(Γ). Moreover we know that H_µ^∞(Γ) is Γ-injective. Thus it suffices to apply:

Exercice. Let X ,→Y be an embedding,Z an injective object. Assume that there exists a unique morphim Y →Z, which is moreover an embedding. Then X ,→Y is essential.

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Boundary actions The Gromov boundary ofFO_Q

The Gromov boundary of F O

_Q

Classical case: free groupΓ= Γ =F_N.

Word length: |g|, spheres: S_n={g ∈F_N;|g|=n}.

Gromov boundary ∂GFN: set of infinite reduced words.

The topology of the compactification β_GF_N =F_Nt∂_GF_N can be described by specifying the unital sub-C^∗-algebraC(β_GF_N)⊂`^∞(F_N):

C(β_GF_N) =S

mC(β_GF_N)_m where

C(β_GF_N)_m ={f ∈`^∞(F_N)|f depends only on firstm letters}

={(f_k)_k ∈L`^∞

k C(S_k)| ∀k ≥m f_k+1=f_k◦ρ_k} where ρk :Sk+1→Sk “forgets last letter”.

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Boundary actions The Gromov boundary ofFO_Q

The Gromov boundary of F O

_Q

The topology of the compactification βGFN =FNt∂GFN can be described by specifying the unital sub-C^∗-algebraC(β_GF_N)⊂`^∞(F_N):

C(βGFN) =S

mC(βGFN)m where

C(β_GF_N)_m ={f ∈`^∞(F_N)|f depends only on firstm letters}

={(f_k)_k ∈L`^∞

k C(S_k)| ∀k ≥m f_k+1=f_k◦ρ_k} where ρ_k :S_k+1→S_k “forgets last letter”.

Quantum case: Γ=FOQ,N ≥3.

Recall `^∞(Γ) =L`^∞

k≥0B(H_k) and we have canonical isometries V_k :H_k₊₁→H_k⊗H₁ from the Temperley-Lieb category.

Theorem (Vaes-Vergnioux ’05)

Put C(β_GFO_Q)m={(f_k)_k | ∀k ≥m f_k+1 =V_k^∗(f_k ⊗id)V_k}. Then C(β_GFO_Q) =S

mC(β_GFO_Q)m is a sub-FO_Q-C^∗-algebra of`^∞(FO_Q).

We also denote C(∂_GFO_Q) =C(β_GFO_Q)/c₀(FO_Q), which is still a unital

∗

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Boundary actions AnFO_Q-boundary

An F O

_Q

-boundary

We have “categorical traces” qtr_k :B(H_k)→C. They satisfyqtr_k+1(V_k^∗(a⊗id)V_k) =qtr_k(a)

Iwe get a stateω= lim−→qtr_k onC(∂_GFO_Q).

One checks that ω is µ-stationary for µ=qtr₁∈B(H₁)^∗⊂c₀(FO_Q)^∗. Denote C_r(∂_GFO_Q) the image of the GNS representation of ω.

Theorem (Vaes-Vergnioux ’05)

Assume N ≥3. Then P_ω extends to a normal ∗-isomorphism P_ω:C_r(∂FO_Q)⁰⁰→H_µ^∞(FO_Q).

Theorem (KKSV ’20)

For N ≥3,ω is the uniqueµ-stationary state on C(∂_GFO_Q).

Hence Cr(∂GFOQ) is anFOQ-boundary. ^proof For N= 2, FO_Q is amenable, the only FO_Q-boundary isC.

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Applications

Outline

An FO_Q-boundary

3 Applications

Uniqueness of trace

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Applications Uniqueness of trace

Uniqueness of trace

Assume that Γacts faithfully on someΓ-boundary A. Then:

ifΓ is unimodular, h is the unique trace on C_red^∗ (Γ) ;

else C_red^∗ (Γ) does not admit any KMS state wrt the scaling group.

Question: in the unimodular case, does uniqueness of trace imply the existence of a faithful boundary action?

For N ≥3,FOQ acts faithfully on ∂GFOQ.

Note: in this case, uniqueness of trace was already proved in [VV ’05]. In the non-unimodular case, the absence ofτ-KMS state is new.

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Applications Universal boundary and the amenable radical

Universal boundary and the amenable radical

Recall that an injective envelope is an injective and essential extension.

Theorem (Hamana, KKSV ’20)

C admits an injective envelope C(∂FΓ) :=IΓ(C), which is unique up to unique isomorphism. We call it the Furstenberg boundary of Γ.

Then anyΓ-boundary embeds in a unique way in C(∂_FΓ).

There exists a faithfulΓ-boundary iff Γy∂_FΓ is faithful.

In the classical case the kernel of this action is the maximal amenable normal subgroup of Γ (amenable radical).

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Applications Universal boundary and the amenable radical

Universal boundary and the amenable radical

Theorem (Hamana, KKSV ’20)

C admits an injective envelope C(∂_FΓ) :=I_Γ(C), which is unique up to unique isomorphism. We call it the Furstenberg boundary of Γ.

There exists a faithfulΓ-boundary iff Γy∂_FΓ is faithful.

In the classical case the kernel of this action is the maximal amenable normal subgroup of Γ (amenable radical).

M ⊂`^∞(Γ) is called a Baaj-Vaes subalgebra if ∆(M)⊂M⊗M¯ . It is relatively amenable if there exists a UCP Γ-mapT :`^∞(Γ)→M.

The cokernel NF of Γy∂FΓis the unique minimal relatively amenable Baaj-Vaes subalgebra of `^∞(Γ).

Hence there exists a faithful Γ-boundaryiff `^∞(Γ) has no proper relatively amenable Baaj-Vaes subalgebra.

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Proof of unique stationarity for F

_N

S_n ⊂F_N: reduced words of lengthn. µ_n: uniform proba measure onS_n. Gromov boundary: ∂GFN 'S∞. PutXg ={g· · · reduced} ⊂S∞. Proposition

Let ω be a proba measure on S∞ such thatµ1∗ω=ω. Then for any g ∈F_N we haveω(X_g) = (#S_|g_|)⁻¹.

Observe that the assumption implies µ^∗k∗ω =ω andµ_k∗ω=ω for all k.

It is sufficient to prove lim_n(µ_n∗ω)(X_g)≤(#S_|g_|)⁻¹: indeed both sides sum up to 1 when |g|is fixed.

We have (µn∗ω)(Xg) = (#Sn)⁻¹P

|h|=nω(hXg).

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Proof of unique stationarity for F

_N

It is sufficient to prove lim_n(µ_n∗ω)(X_g)≤(#S_|g_|)⁻¹: indeed both sides sum up to 1 when |g|is fixed.

We have (µn∗ω)(Xg) = (#Sn)⁻¹P

|h|=nω(hXg).

Case 1: the last letter of g is not simplified in the producthg, i.e.

|hg|=|g|+n−2l with 0≤l ≤ |g| −1. Then hX_g =X_hg and when l is fixed these subsets are pairwise disjoint. Hence for fixedl:

P{ω(hX_g);|h|=n,|hg|=|g|+n−2l} ≤1.

Case 2: use the trivial estimateω(hX_g)≤1. In this case the last |g|letters of h are fixed, equal to g⁻¹, so we have (2N−1)^n−|g^|such elements h.

Altogether (µn∗ω)(Xg) ≤(#Sn)⁻¹P|g|−1

l=0 1 + (#Sn)⁻¹(2N−1)^n−|g^|

= (#S_n)⁻¹|g|+ (#S_|g_|)⁻¹ →_n∞(#S_|g|)⁻¹.

Indeed #S_n= 2N(2N−1)ⁿ⁻¹. ^back