Maximal rigid subalgebras of deformations and
L2-cohomology

Santiago, Rolando de; Hayes, Ben; Hoff, Daniel J.; Sinclair, Thomas

doi:10.2140/apde.2021.14.2269

Cited by 4 publications

(8 citation statements)

References 36 publications

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“…By using [Ioa12, Lemma 9.2(1)] we get , and by assumption we must have . Hence, in combination with being -rigid and the fact that is a mixing -bimodule relative to , we get from [dSHHS20, Corollary 6.7] that is -rigid.…”

Section: Malleable Deformations For Von Neumann Algebras: Classmentioning

confidence: 96%

“…This notion has been successfully used in the framework of his deformation/rigidity theory and led to a plethora of remarkable results in the theory of von Neumann algebras, see the surveys [Pop07a,Vae10,Ioa14,Ioa18]. We also refer the reader to [dSHHS20] for recent developments on s-malleable deformations.…”

Section: Malleable Deformationsmentioning

confidence: 99%

“…Note that (3.3) gives, in particular, that , for any . Therefore, we may apply [dSHHS20, Theorem 3.5] (see also [Dri21, Theorem 3.2]) to deduce that . Using [dSHHS20, Proposition 5.6] there exists a non-zero projection such that .…”

Section: Malleable Deformations For Von Neumann Algebras: Classmentioning

confidence: 99%

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Measure equivalence rigidity via s-malleable deformations

Drimbe¹

2023

Compositio Math.

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We single out a large class of groups ${\rm {\boldsymbol {\mathscr {M}}}}$ for which the following unique prime factorization result holds: if $\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}}$ and $\Gamma _1\times \cdots \times \Gamma _n$ is measure equivalent to a product $\Lambda _1\times \cdots \times \Lambda _m$ of infinite icc groups, then $n \ge m$ , and if $n = m$ , then, after permutation of the indices, $\Gamma _i$ is measure equivalent to $\Lambda _i$ , for all $1\leq i\leq n$ . This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Class ${\rm {\boldsymbol {\mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $\Gamma$ for which either (i) $\Gamma$ is an arbitrary wreath product group with amenable base or (ii) $\Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Finally, for groups $\Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(\Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].

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Section: Malleable Deformations For Von Neumann Algebras: Classmentioning

confidence: 96%

Section: Malleable Deformationsmentioning

confidence: 99%

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Measure equivalence rigidity via s-malleable deformations

Drimbe¹

2023

Compositio Math.

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“…Here, we denoted by M I = ⊗i∈I M i . Next, by using some general recent results on s-malleable deformations [dSHHS20] and an augmentation technique developed in [CD-AD20] we are able to use (1.2) and (1.3) to show that there exists a non-zero projection…”

Section: L(γmentioning

confidence: 99%

“…In the framework of his powerful deformation/rigidity techniques, this notion has led to a remarkable progress in the theory of von Neumann algebras, see the surveys [Po07,Va10a,Io12b,Io17]. See also [dSHHS20] for a comprehensive overview on s-malleable deformations and for recent developments.…”

Section: Two Classes Of Von Neumann Algebras That Admit S-malleable D...mentioning

confidence: 99%

Product rigidity in von Neumann and C$^*$-algebras via s-malleable deformations

Drimbe

2020

Preprint

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We provide a new large class of countable icc groups A for which the product rigidity result from [CdSS15] holds: if Γ1, . . . , Γn ∈ A and Λ is any group such that L(Γ1 ×• • •×Γn) ∼ = L(Λ), then there exists a product decomposition Λ = Λ1 × • • • × Λn such that L(Λi) is stably isomorphic to L(Γi), for any 1 ≤ i ≤ n. Class A consists of groups Γ for which L(Γ) admits an s-malleable deformation and it includes all non-amenable groups Γ such that either (a) Γ admits an unbounded 1-cocycle into its left regular representation, or (b) Γ is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W * -superrigid groups and new rigidity results in the C * -algebra theory.

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