Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups

Boutonnet, Rémi; Carderi, Alessandro

doi:10.48550/arxiv.1310.5864

Cited by 2 publications

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“…Hyperbolic groups. As a first application of our criterion, we give a new proof of the main result of [BC14]. Note that one can also recover the results from [BC14] about relatively hyperbolic groups.…”

Section: Examplesmentioning

confidence: 66%

“…We have already used this lemma in [BC14] without giving a proof. Let us give a proof for completeness.…”

Section: Proposition 21 ([Bc14]mentioning

confidence: 99%

“…In [Ho14], C. Houdayer provided uncountably many non-isomorphic examples of abelian maximal amenable subalgebras in II 1 -factors. Last year the authors showed in [BC14] that any infinite maximal amenable subgroup in a hyperbolic group Γ gives rise to a maximal amenable von Neumann subalgebra of LΓ. Note that hyperbolic groups can have property (T).…”

Section: Introductionmentioning

confidence: 99%

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Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

Boutonnet

Carderi

2015

Geom. Funct. Anal.

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Abstract. We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms of Λ-invariant measures on some compact Γ-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C * -algebra.

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Section: Examplesmentioning

confidence: 66%

“…We have already used this lemma in [BC14] without giving a proof. Let us give a proof for completeness.…”

Section: Proposition 21 ([Bc14]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

Boutonnet

Carderi

2015

Geom. Funct. Anal.

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“…Recently, the author vastly generalized in [Ho12a,Ho12b] Popa's results from [Po83] and obtained many new examples of maximal amenable subalgebras inside the crossed product II 1 factors associated with free Bogoljubov actions of amenable groups. Very recently, Boutonnet and Carderi showed in [BC13] that any infinite maximal amenable subgroup Λ in a Gromov word-hyperbolic group Γ gives rise to a maximal amenable subalgebra L(Λ) inside the group von Neumann algebra L(Γ). For other related results regarding maximal amenability in the framework of II 1 factors, we refer the reader to [Br12,Fa06,Ga09,Ge95,Jo10,Po13,Sh05].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Gamma Stability in Free Product von Neumann Algebras

Houdayer

2014

Commun. Math. Phys.

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Abstract. Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer M ϕ 1 1 is diffuse. We first show that any intermediate subalgebra M1 ⊂ Q ⊂ M which has nontrivial central sequences in M is necessarily equal to M1. Then we obtain a general structural result for all the intermediate subalgebras M1 ⊂ Q ⊂ M with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III1 factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras. Introduction and statement of the main resultsA von Neumann algebra M ⊂ B(H) (with separable predual) is amenable if there exists a norm one projection E : B(H) → M . By Connes' celebrated result [Co75b], all the amenable von Neumann algebras are hyperfinite. Moreover, the amenable or hyperfinite factors are completely classified by their flows of weights (see [Co72,Co75b,Co85,Ha84] Since the amenable von Neumann algebras form a monotone class, any von Neumann algebra admits maximal amenable subalgebras. The first concrete examples of maximal amenable subalgebras inside II 1 factors were obtained by Popa in [Po83]. He showed that any generator masa A in a free group factor L(F n ) with n ≥ 2 is maximal amenable. This result answered in the negative a question raised by Kadison. Indeed, A ⊂ L(F n ) is an abelian subalgebra generated by a selfadjoint operator and yet there is no intermediate hyperfinite subfactor in L(F n ) which contains A as a subalgebra. Popa discovered in [Po83] a powerful method to prove that a given amenable subalgebra is maximal amenable inside an ambient II 1 factor. Using this strategy for the generator masa A ⊂ L(F n ), he first showed that A satisfies a certain asymptotic orthogonality property and then deduced that A is maximal amenable in L(F n ) using various mixing techniques. His results actually showed that the generator masa A is maximal Gamma inside L(F n ). Recall that a II 1 factor M (with separable predual) has property Gamma of Murray and von Neumann [MvN43] if there exists a sequence of unitaries u n ∈ U (M ) such that lim n→∞ τ (u n ) = 0 and lim n→∞ xu n − u n x 2 = 0 for all x ∈ M . Subsequently, Cameron, Fang, Ravichandran and White proved in [CFRW08] that the radial masa in a free group factor L(F n ) with 2 ≤ n < ∞ is maximal amenable. Recently, the author vastly generalized in [Ho12a, Ho12b] Popa's results from [Po83] and obtained many new examples of maximal amenable subalgebras inside the crossed product II 1 factors associated with free Bogoljubov actions of amenable groups. Very recently, Boutonnet and Carderi showed in [BC13] that any infinite maximal amenable subgroup Λ in a Gromov word-hyperbolic group Γ gives rise to a maximal amenable subalgebra L(Λ) inside the group von...

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Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups

Cited by 2 publications

References 0 publications

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

Gamma Stability in Free Product von Neumann Algebras

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