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

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We show that if H ≤ 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.