  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.
