Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

Houdayer, Cyril; Ricard, Éric

doi:10.1016/j.aim.2011.06.010

Cited by 48 publications

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References 56 publications

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“…Combining Corollary 3.2 with the deformation/rigidity theorems for free Araki-Woods factors and for free product factors obtained in [HR10,HU15], we get the following theorem.…”

Section: Proofs Of Theorem a And Corollaries B C Dmentioning

confidence: 78%

Classification of a family of non-almost-periodic free Araki–Woods factors

2019

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We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors Γ(µ, m) ′′ up to isomorphism. We do this by showing that free Araki-Woods factors Γ(µ, m) ′′ arising from finite symmetric Borel measures µ on R whose atomic part µa is nonzero and not concentrated on {0} have the joint measure class C( k≥1 µ * k ) as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.2010 Mathematics Subject Classification. 46L10, 46L54, 46L36.1 2 CYRIL HOUDAYER, DIMITRI SHLYAKHTENKO, AND STEFAAN VAES µ c * µ c ≺ µ c . We find in particular that in that case, the free Araki-Woods factor does not depend on the multiplicity function m. But we also show that in other cases, Γ(µ, m) ′′ does depend on m.In order to state our main results, we first introduce some terminology. For every σ-finite Borel measure µ on R, we denote by C(µ) the measure class of µ, defined as the set of all Borel sets U ⊂ R with µ(U ) = 0. Note that C(µ) = C(ν) if and only if µ ∼ ν, while C(µ) ⊂ C(ν) if and only if ν ≺ µ. For any sequence of measures (µ k ) k∈N , we denote by k∈N µ k any measure with the property that C( k∈N µ k ) = k∈N C(µ k ). We denote by µ = µ c + µ a the unique decomposition of a measure µ as the sum of a continuous and an atomic measure.We show that free Araki-Woods factors Γ(µ, m) ′′ arising from finite symmetric Borel measures µ on R whose atomic part µ a is nonzero and not concentrated on {0} have the joint measure class C( k≥1 µ * k ) as an invariant. More precisely, we obtain the following result.Theorem A. Let µ, ν be finite symmetric Borel measures on R and m, n : R → N ∪ {+∞} symmetric Borel multiplicity functions. Assume that ν has at least one atom not equal to 0.If the free Araki-Woods factors Γ(µ, m) ′′ and Γ(ν, n) ′′ are isomorphic, then there exists an isomorphism that preserves the free quasi-free states. In particular, the joint measure classes C( k≥1 µ * k ) and C( k≥1 ν * k ) are equal.Denote by S(R) the set of all finite symmetric Borel measures µ = µ c + µ a on R satisfying the following two properties:Denote by Λ(µ a ) the countable subgroup of R generated by the atoms of µ a and by δ Λ(µa) a finite atomic measure on R whose set of atoms equals Λ(µ a ).Combining our Theorem A with the isomorphism Theorem 4.1 below, we obtain a complete classification of the free Araki-Woods factors arising from measures in S(R). Here and elsewhere in this paper, we call isomorphism between von Neumann algebras M and N any bijective * -isomorphism. Even when M and N are equipped with distinguished faithful normal states, isomorphisms are not assumed to preserve these states. Corollary B. The set of free Araki-Woods factors Γ(µ, m) ′′ : µ ∈ S(R) and m : R → N ∪ {+∞} is a symmetric Borel multiplicity function is exactly classified, up to isomorphism, by the countable subgroup Λ(µ a ) and the measure class C(µ c * δ Λ(µa) )....

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“…Combining Corollary 3.2 with the deformation/rigidity theorems for free Araki-Woods factors and for free product factors obtained in [HR10,HU15], we get the following theorem.…”

Section: Proofs Of Theorem a And Corollaries B C Dmentioning

confidence: 78%

Classification of a family of non-almost-periodic free Araki–Woods factors

2019

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“…Combining [HR10, Theorem A] and [AD93, Lemma 4.6 and Theorem 4.9], the type II 1 factor qMq has the complete metric approximation property. Moreover, since L(G) ⊗ N is amenable, the qMq-qMq-bimodule L 2 (q Mq) ⊖ L 2 (qMq) is weakly contained in the coarse bimodule L 2 (qMq) ⊗ L 2 (qMq) (see [HR10,Section 4] Since π is faithful and mixing, Theorem D implies that M is a strongly solid factor. Let (t, n) ∈ R × Z be any element such that U t ⊗ π n = 1.…”

Section: Moreover We Havementioning

confidence: 99%

Asymptotic structure of free Araki–Woods factors

Houdayer¹,

Raum²

2015

Math. Ann.

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“…We note that the first solution to the Cartan subalgebra problem for type III factors in our framework was obtained by Houdayer and Ricard [HR10]. They followed the proof of [OP07] by exploiting techniques in [CH08], that is, the use of Popa's deformation and intertwining techniques together with the continuous core decomposition.…”

Section: Introductionmentioning

confidence: 99%

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Isono

2019

Analysis & PDE

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Let G be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor N ⊂ L ∞ (G), which is an image of a faithful normal conditional expectation, and for any σ-finite factor B, the tensor product N ⊗ B has no Cartan subalgebras. This generalizes our previous work that provides the same result when B is finite. In the proof, we establish Ozawa-Popa and Popa-Vaes's weakly compact action on the continuous core of L ∞ (G) ⊗ B as the one relative to B, by using an operator valued weight to B and the central weak amenability of G.We first recall some notions in Tomita-Takesaki theory. We refer the reader to [Ta01] for this theory, and to [Ha77a, Ha77b] and [Ta01, Chapter IX. §4] for operator valued weights.Let M be a von Neumann algebra and ϕ a faithful normal semifinite weight on M . Put n ϕ := {x ∈ M | ϕ(x * x) < ∞} and denote by Λ ϕ : n ϕ → L 2 (M, ϕ) the canonical embedding. We denote the modular operator, modular conjugation, and modular action for M ⊂ B(L 2 (M, ϕ)) by ∆ ϕ , J ϕ and σ ϕ respectively. The Hilbert space L 2 (M, ϕ) with J ϕ and with its positive cone P ϕ is called the standard representation for M [Ta01, Chapter IX. §1], which does not depends on the choice of ϕ. Any state on M is represented by a vector state, from which the vector is uniquely chosen from P ϕ . Any element α ∈ Aut(M ) is written as α = Ad u by a unique u ∈ B(L 2 (M, ϕ)) which preserves the standard representation structure. The crossed product M ⋊ σ ϕ R by the modular action is called the continuous core [Ta01, Chapter XII. §1] and is written as C ϕ (M ), which is equipped with the dual

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Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

Abstract: International audienc

Cited by 48 publications

References 56 publications

Classification of a family of non-almost-periodic free Araki–Woods factors

Classification of a family of non-almost-periodic free Araki–Woods factors

Asymptotic structure of free Araki–Woods factors

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

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