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 Chifan et al. (Geom Funct Anal 26(1): 136–159, 2016) holds: if $\Gamma _1,\ldots ,\Gamma _n\in {\mathcal {A}}$ 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 $1\le i\le n$. 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$^*$-superrigid groups and new rigidity results in the C$^*$-algebra theory.