Product rigidity in von Neumann and C$^*$-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 $L(\Gamma_1\times\dots\times\Gamma_n)\cong L(\Lambda)$, then there exists a product decomposition $\Lambda=\Lambda_1\times\dots\times \Lambda_n$ such that $L(\Lambda_i)$ is stably isomorphic to $L(\Gamma_i)$, for any $1\leq i\leq n$. 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$^*$-superrigid groups and new rigidity results in the C$^*$-algebra theory.