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

TL;DR

This paper proves finiteness results for invariant subalgebras of group von Neumann and $C^*$-algebras for lattices in higher rank Lie groups, revealing their structure is governed by normal subgroups.

Contribution

It establishes that invariant subalgebras are generated by normal subgroups and characterizes equivariant conditional expectations in this setting.

Findings

Invariant von Neumann subalgebras are generated by normal subgroups.

Invariant $C^*$-subalgebras correspond to normal subgroups.

Conditional expectations are canonical onto subalgebras generated by normal subgroups.

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.