Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations

Abstract

We provide a new large class of countable icc groups $\mathcal{A}$ for which the product rigidity result from [CdSS15] holds: if ${\Gamma }_1,\dots ,{\Gamma }_n\in \mathcal{A}$ and $\Lambda$ is any group such that $L({\Gamma }_1\times \cdots \times {\Gamma }_n)\cong L(\Lambda )$, then there exists a product decomposition $\Lambda ={\Lambda }_1\times \cdots \times {\Lambda }_n$ such that $L({\Lambda }_i)$ is stably isomorphic to $L({\Gamma }_i)$, for any $1\le i\le n$. Class $\mathcal{A}$ consists of groups $\Gamma$ for which $L(\Gamma )$ admits an s-malleable deformation in the sense of Sorin Popa and it includes all non-amenable groups $\Gamma$ such that either (a) $\Gamma$ admits an unbounded 1-cocycle into its left regular representation, or (b) $\Gamma$ is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W${}^{\ast }$-superrigid groups and new rigidity results in the C${}^{\ast }$-algebra theory.

arXiv commentTo appear as such in Communications in Mathematical Physics

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