Paper detail

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

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