The $C^*$--ISR Property for $\text{PSL}_n(\mathbb{Z})$

Abstract

A discrete group $\Gamma$ has the \emph{$C^*$--ISR property} if every unital $\Gamma$--invariant $C^*$--subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$--ISR property for every $n\geq3$.