Asymptotic structure of free Araki–Woods factors

Houdayer, Cyril; Raum, Sven

doi:10.1007/s00208-015-1168-1

Cited by 20 publications

(16 citation statements)

References 46 publications

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“…In this respect, our approach is similar to the one we developed in our previous paper [HI15]. We refer the reader to [HR14,HU15,HV12,Is12,Is13] for other structural/rigidity results for type III factors involving the continuous core decomposition.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 96%

“…Following [HR14,Oz04], we say that a von Neumann algebra M is ω-semisolid if for any von Neumann subalgebra N ⊂ M with expectation such that the relative commutant N ′ ∩ M ω has no type I direct summand, we have that N is amenable. The next corollary strengthens the indecomposability properties of crossed product von Neumann algebras B ⋊ Γ arising from arbitrary actions Γ B of bi-exact discrete groups on abelian von Neumann algebras (see [Oz04,HV12,Is12] for previous results).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In [HR14], the first named author and Raum investigated the asymptotic structure of Shlyakhtenko's free Araki-Woods factors [Sh96]. Among other things, they proved in [HR14, Theorem A] that any diffuse von Neumann algebra M with separable predual satisfying Ozawa's condition (AO) is ω-solid, that is, for any von Neumann subalgebra with expectation N ⊂ M such that the relative commutant N ′ ∩ M ω is diffuse, we have that N is amenable.…”

Section: Further Remarksmentioning

confidence: 99%

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Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

Houdayer

Isono

2016

Commun. Math. Phys.

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Abstract. We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras M = B ⋊Γ arising from arbitrary actions Γ B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra N ′ ∩ M ω of any nonamenable von Neumann subalgebra with normal expectation N ⊂ M . We use this result to show that for any strongly ergodic essentially free nonsingular action Γ (X, µ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor L ∞ (X) ⋊ Γ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every 0 < λ ≤ 1, a type III λ strongly ergodic essentially free nonsingular action F∞ (X λ , µ λ ) of the free group F∞ on a standard probability space so that the corresponding group measure space type III λ factor L ∞ (X λ , µ λ ) ⋊ F∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type III factors with no nontrivial central sequence.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 96%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Further Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

Houdayer

Isono

2016

Commun. Math. Phys.

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“…Since Q ⊂ qM q is diffuse and with expectation and since M is solid (see [HR14, Theorem A] and [HI15, Theorem 7.1] which does not require separability of the predual), the relative commutant Q ′ ∩ qM q is amenable. Up to replacing Q by Q ∨ Q ′ ∩ qM q which is still amenable and globally invariant under the modular automorphism group σ ϕq , we may assume that Q ′ ∩ qM q = Z(Q).…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Structure of modular invariant subalgebras in free Araki–Woods factors

Boutonnet

Houdayer

2016

Anal. PDE

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“…Instead, we use a simple analysis of central sequences in the presence of weak mixing property (see Lemma 6). This type of analysis, whose prototype appears in an old work of Popa [22] on type II 1 factors, was used by Houdayer-Raum [13] and Boutonnet-Houdayer [6] in combining with the weak mixing property for some questions on free Araki-Woods factors. In this respect, the novelty of this work is the use of a bounded projection onto the span of 'letters' (see equation (1) and Lemma 5) in such an analysis.…”

Section: Introductionmentioning

confidence: 99%