On invariant subalgebras of group $C^*$ and von Neumann algebras

Abstract: Given an irreducible lattice $\Gamma$ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma$-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal{L}(\Gamma)$, and for the $\Gamma$-invariant unital $C^*$-subalgebras of the reduced group $C^*$-algebra $C^*_{\rm red}(\Gamma)$. We use these results to show that: (i) every $\Gamma$-invariant von Neumann subalgebra of $\mathcal{L}(\Gamma)$ is generated by a normal subgroup; and (ii) given a non-amenable unitary representation $\pi$ of $\Gamma$, every $\Gamma$-equivariant conditional expectation on $C^*_\pi(\Gamma)$ is the canonical conditional expectation onto the $C^*$-subalgebra generated by a normal subgroup.