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

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