On invariant subalgebras of group and von Neumann algebras

Abstract

AbstractGiven 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^*_{\mathrm {red}}(\Gamma )$. We use these results to show that: (i) every$\Gamma $-invariant von Neumann subalgebra of$\mathcal {L}(\Gamma )$is generated by a normal subgroup; and (ii) given a weakly mixing unitary representation$\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.