Rigidity results for group von Neumann algebras with diffuse center

Abstract

We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. Then whenever $H$ is an \emph{arbitrary} group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal{L}(H)$ it must be the case that $H=B\times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal{L}(G)$ and $\mathcal{L}(H)$. This yields new applications to the classification of group C${}^{\ast }$-algebras, including examples of non-amenable groups which are recoverable from their reduced C${}^{\ast }$-algebras but not from their von Neumann algebras.

v3: typos corrected and some results are updated to hfc radical