College of Liberal Arts & Sciences

# Operator Theory Seminar

**Abstract:**

We provide a fairly large family of amalgamated free product groups $\Gamma=\Gamma_1\ast_{\Sigma}\Gamma_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that $\Gamma_i$ is a product of two icc non-amenable bi-exact groups, and $\Sigma$ is icc amenable with trivial one-sided commensurator in $\Gamma_i$, for every $i=1,2$. Then $\Gamma$ satisfies the following rigidity property: *any* group $\Lambda$ such that $L(\Lambda)$ is isomorphicÂ to $L(\Gamma)$ admits an amalgamated free product decomposition $\Lambda=\Lambda_1\ast_\Delta \Lambda_2$ such that the inclusions $L(\Delta)\subseteq L(\Lambda_i)$ and $L(\Sigma)\subseteq L(\Gamma_i)$ are isomorphic, for every $i=1,2$. This result significantly strengthens some of the previous Bass-Serre rigidity results for von Neumann algebras.

As a corollary, we obtain the first examples of amalgamated free product groups which are $W^*$-superrigid. In addition this leads to the first examples of *non virtually* abelian group that can be completely reconstructed from their reduced $C^*$-algebra. This is based on a recent joint work with Adrian Ioana.