Cocompact amenable closed subgroups: weakly inequivalent representations in the left-regular representation

TL;DR

This paper demonstrates that for certain groups with cocompact amenable subgroups, the reduced group C*-algebra is not simple, revealing new insights into the structure of unitary representations and their relation to the left-regular representation.

Contribution

It establishes a link between cocompact amenable subgroups and the non-simplicity of the reduced group C*-algebra, highlighting the existence of weakly contained but not weakly equivalent representations.

Findings

Reduced group C*-algebra of G is not simple under given conditions

Existence of unitary representations weakly contained but not equivalent to the left-regular representation

Open problem on constructing non-discrete topologically simple groups with specific properties

Abstract

We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.