Operator Theory Seminar

September 12, 2017 - 1:30pm
309 VAN

“New Examples of W^*-Super Rigid Groups (Cont.)” by Professor Ionut Chifan, Department of Mathematics, The University of Iowa

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: \emph{any} group $\La$  such that $L(\La)$ is isomorphic  to $L(\G)$ admits an amalgamated free product decomposition $\La=\La_1\ast_\Delta \La_2$ such that the inclusions $L(\Delta)\subseteq L(\La_i)$ and $L(\Sigma)\subseteq L(\G_i)$ 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