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

A discrete group $\mathrm{\Gamma }$ has the \emph{$C^{\ast }$–ISR property} if every unital $\mathrm{\Gamma }$–invariant $C^{\ast }$–subalgebra $A\subseteq C_r^{\ast }(\mathrm{\Gamma })$ is of the form $C_r^{\ast }(N)$ for a normal subgroup $N◃\mathrm{\Gamma }$. We show that ${\text{PSL}}_n(\mathbb{Z})$ satisfies the $C^{\ast }$–ISR property for every $n\ge 3$.