The radial MASA in free orthogonal quantum groups

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The radial MASA in free orthogonal quantum groups

Roland Vergnioux joint work with Amaury Freslon

University of Normandy (France)

Greifswald, July 11th, 2016

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Discrete Quantum Groups

Outline

1 Discrete Quantum Groups WoronowiczC^∗-algebras

The Haar measure and corepresentations

2 The radial subalgebra in C^∗(FON) Free orthogonal quantum groups The radial subalgebra

3 The classical case

Maximal abelian subalgebras The radial MASA in L(F_N)

4 About the proof The main estimate The main lemma

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Discrete Quantum Groups WoronowiczC^∗-algebras

From classical to quantum

Γ discrete group IC_red^∗ (Γ)⊂B(`²Γ) Elements g ∈γ Iunitariesλ(g)∈C_red^∗ (Γ)

Additional structure to recover Γ from C_red^∗ (Γ): coproduct

∆ :C_red^∗ (Γ)→C_red^∗ (Γ)⊗C_red^∗ (Γ), λ(g)7→λ(g)⊗λ(g).

Then Γ ={u ∈U(C_red^∗ (Γ))|∆(u) =u⊗u}.

Definition

Woronowicz C^∗-algebra : unitalC^∗-algebra Aand unital ∗-homomorphism

∆ :A→A⊗Asuch that (∆⊗id)∆ = (id⊗∆)∆,

∆(A)(A⊗1) and ∆(A)(1⊗A) dense in A⊗A.

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From classical to quantum

Definition

Woronowicz C^∗-algebra : unital C^∗-algebra Aand unital ∗-homomorphism

Example: A=C_red^∗ (Γ) but alsoC_max^∗ (Γ).

Igeneral notationA=C^∗(Γ),Γ“discrete quantum group”.

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From classical to quantum

Definition

Woronowicz C^∗-algebra : unital C^∗-algebra Aand unital ∗-homomorphism

Example: A=C(G), G a (classical) compact group, with

∆ :C(G)→C(G ×G), ∆(f) = ((g,h)7→f(gh)).

Also A=C(Gq),q-deformation a simple compact Lie groupG.

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Discrete Quantum Groups The Haar measure and corepresentations

General facts

Theorem (Woronowicz, the “Haar state”) There exists a unique state h∈C^∗(Γ)^∗ such that

(h⊗id)∆ = (id⊗h)∆ = 1·h.

IGNS representationλ:C^∗(Γ)C_red^∗ (Γ)⊂B(`²Γ).

IVon Neumann algebraL(Γ) =C_red^∗ (Γ)⁰⁰⊂B(`²Γ).

Corepresentation: u ∈C^∗(Γ)⊗B(H_u) unitary s.t. (∆⊗id)(u) =u13u23. Can define: direct sums, tensor products, contragredient, intertwiners, ...

IC^∗-tensor categoryCorep(Γ), irreducible corepsIrr(Γ). Theorem (“Peter-Weyl-Woronowicz”)

Every corepresentation is equivalent to a direct sum of irreducibles. The coefficients u_ζ,ξ= (id⊗ω_ζ,ξ)(u) of irreducible corepresentations span a dense subspace C[Γ]⊂C^∗(Γ).

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Discrete Quantum Groups The Haar measure and corepresentations

General facts

Theorem (Woronowicz, the “Haar state”) There exists a unique state h∈C^∗(Γ)^∗ such that

(h⊗id)∆ = (id⊗h)∆ = 1·h.

IGNS representationλ:C^∗(Γ)C_red^∗ (Γ)⊂B(`²Γ).

IVon Neumann algebraL(Γ) =C_red^∗ (Γ)⁰⁰⊂B(`²Γ).

Corepresentation: u ∈C^∗(Γ)⊗B(H_u) unitary s.t. (∆⊗id)(u) =u13u23. Can define: direct sums, tensor products, contragredient, intertwiners, ...

IC^∗-tensor categoryCorep(Γ), irreducible corepsIrr(Γ).

Theorem (“Peter-Weyl-Woronowicz”)

Every corepresentation is equivalent to a direct sum of irreducibles.

The coefficients u_ζ,ξ= (id⊗ω_ζ,ξ)(u) of irreducible corepresentations span a dense subspace C[Γ]⊂C^∗(Γ).

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The radial subalgebra inC^∗(FO_N)

Outline

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The radial subalgebra inC^∗(FO_N) Free orthogonal quantum groups

Free orthogonal quantum groups

Definition (Wang)

The orthogonal free quantum group FON is given by

C^∗(FO_N) =hv_ij,1≤i,j ≤N |v_ij^∗ =vij,v = (vij)ij unitaryi and the coproduct ∆(v_ij) =P

v_ik⊗v_kl.

There are surjective morphisms of Woronowicz C^∗-algebras C^∗(FO_N)C^∗(Z^∗N₂ ),v_ij 7→δ_ija_i C^∗(FO_N)C(O_N), v_ij 7→(g 7→g_ij) Known results aboutL(FON),N ≥3:

it is a fullII₁ factor [Vaes-V.] with Property AO [V., Vaes-V.]

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

it is stronly solid [Isono, Fima-V.] hence has no regular MASA it embeds in R^ω [Brannan-Collins-V.]

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The radial subalgebra inC^∗(FO_N) The radial subalgebra

The radial subalgebra

Character of a corepresentation: χ_u= (id⊗Tr)(u)∈C^∗(Γ).

Definition

The radial subalgebra is B=λ(χv)⁰⁰ ⊂L(FO_N).

One can also consider B =C^∗(χ_v)⊂C^∗(FO_N).

Known facts:

Sp(χ_v) = [−N,N] andB 'C([−N,N]).

Brannan: the orthogonal projection extends to a cond. expectation E :C^∗(FO_N)→B. The positive forms ev_t◦E arec₀ and converge to εast →N IHaagerup Approximation Property.

Banica: Sp(λ(χ_v)) = [−2,2] and B'L^∞([−2,2],leb).

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The radial subalgebra

Definition

Why radial? Analogy FON ↔ FN =ha_ii,v ↔ a=diag(ai,a⁻¹_i ).

In L(F_N) one hasP

f(g)λ(g)∈χ⁰⁰_a iff f(g) =ϕ(|g|). Known facts:

χ⁰⁰_a is a singular MASA [Pytlik, Radulescu]

∃ ξ⊥χ⁰⁰_a such thatµξ:f ⊗g 7→(ξ|fξg) on (Spχa)² is equivalent to the product measureh⊗h [Dykema-Mukherjee]

Deep result [Voiculescu]: for any MASA C ⊂L(FN) there exists ξ⊥C st µ_ξ is not singular wrt h⊗h.

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The radial subalgebra

Definition

Main result:

Theorem (Freslon-V.)

The subalgebra B⊂L(FON), N ≥3, is a singular MASA. For any ξ ∈C[FO_N]∩B^⊥, the measureµ_ξ on[−2,2]² is equivalent to h⊗h.

Questions: what about the non-Kac caseL(FOQ)?

What about the “generator subalgebras” λ(v_ij)⁰⁰?

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The classical case

Outline

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The classical case Maximal abelian subalgebras

Maximal abelian subalgebras

Terminology

M finite von Neumann algebra with separable predual and fixed traceh B⊂M weakly closed abelian subalgebra, E :M →B cond. exp.

Normalizer: N_M(B) ={u ∈U(M)|u^∗Bu ⊂B}⁰⁰

MASA:B⁰ =B, singular MASA: N_M(B) =B, regular MASA:B⁰=B andN_M(B) =M (Cartan subalgebra), strongly mixing: ∀ (un)n∈U(B)^N stun

→w 0 and∀ x,x⁰ ∈M one haskE(u_n^∗xu_nx⁰)−E(x)E(x⁰)k₂ →0 (SAHP forE)

Elementary facts:

abelian, diffuse and strongly mixing =⇒ singular MASA,

strongly mixing ⇐⇒ kE(xu_nx⁰)k₂ →0 forx (resp. x⁰) spanning a dense subspace in B^⊥ (resp. M)

strongly mixing ⇐=P

|(c_lx|x⁰cl⁰)|² <∞ forx,x⁰ as above and (cl)l

fixed ONB ofL²(B).

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The classical case The radial MASA inL(F_N)

Classical MASAs in L (F

_N

)

Generator MASA B=a⁰⁰

cl =a^l,x =g ∈FN \a^Z,x⁰ =g⁰ ∈FN I(clx|x⁰cl⁰) =δ(g⁰,a^lga^−l⁰).

This is non zero for at most one value of (l,l⁰)! Isingular MASA Radial MASA B= (P

a_i+a_i⁻¹)⁰⁰ cl =dl/kd_lk₂,dl =P

W_lg,Wl ={g ∈FN||g|=l}.

x =g −h with |g|=|h|=k,x⁰ =g⁰ with |g⁰|=k⁰. Then (d_lx|x⁰d_l⁰) = #W_lg ∩g⁰W_l⁰−#W_lh∩g⁰W_l⁰.

Lemma

Denote ws(g,g⁰) = #{x =g⁰· · ·g reduced,|x|=s}.

Then w_s(g,g⁰) =w_s(h,g⁰) + 1/0/−1if |g|=|h|.

I|(c_lx|x⁰c_l⁰)| ≤ min(k,k⁰) + 1

(2N−1)^(l^+l⁰^)/2 ^Isingular MASA

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About the proof

Outline

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About the proof The main estimate

The main estimate

Theorem (Banica)

Irr CorepFO_N ={v^k,k∈N}with v⁰= 1, v¹ =v and v^k⊗v¹'v^k−1⊕v^k+1 'v¹⊗v^k. In particular (χ_k)_k is an ONB of L²(B), where χ_k =χ_vk.

Moreover B^⊥ is spanned by coefficientsv_ζξ^k with ζ⊥ξ ∈H_k =H_vk.

Theorem (Freslon-V.)

For all ζ⊥ξ ∈H_n,ζ⁰,ξ⁰∈H_n⁰ we have

(χlv_ζξⁿ|v_ζⁿ0⁰ξ⁰χl⁰) =O(q^(l+l⁰^)/2) with q ∈]0,1[, q+q⁻¹=N, N≥3.

Istrongly mixing MASA.

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About the proof The main lemma

The main lemma

Partial traces of Jones-Wenzl projections

Fusion rule v^a+b+c ⊂v^a⊗v^b⊗v^c IPa,b,c ∈B(Ha⊗H_b⊗H_c).

Lemma

Xa,b,c = (ida⊗Tr_b⊗id_c)(Pa,b,c) is asymptotically scalar as b → ∞:

X_a,b,c =λ_a,b,c(id_a⊗id_b) +O(λ_a,b,cq^b).

NB: if a orc = 0,Xa,b,c is scalar (intertwiner).

Interpretation: analogy withF_N

v ∈C^∗(FO_N)⊗B(C^N) a=diag(a_i,a⁻¹_i )∈C^∗(F_N)⊗C(W₁) v^k ∈C^∗(FO_N)⊗B(H_k) a_k =P

|g|=kg⊗11_g ∈C^∗(F_N)⊗C(W_k) P_a,b,c ∈B(Ha⊗H_b⊗H_c) 11_W_a+b+c ∈C(Wa×W_b×Wc)

X_a,b,c ∈B(H_a⊗H_b) w_a+b+c(·,·)∈C(W_a×W_b) scalar constant