The Bohr topology of Moore groups

For a locally compact (LC) group G, denote by G+ its underlying group equipped with the topology inherited from its Bohr compactification. G is maximally almost periodic (MAP) if and only if G+ is Hausdorff. If P denotes a topological property, then we say that a MAP group G respects P if G and G+ have the same subspaces with P. In 1962 I. Glicksberg proved that LC Abelian groups respect compactness. We extend this result by showing that LC groups such that all their irreducible unitary representations are finite-dimensional, i.e., [MOORE] groups, do so as well. Moreover, we prove that G equipped with the topology induced by its topological dual is equal to G+ if and only if G belongs to the class [MOORE]. If this is indeed the case, then (a) G additionally respects pseudocompactness, (relative) functional boundedness, and the Lindelöf property, (b) G is connected (respectively zero-dimensional, respectively realcompact) if and only if G+ is connected (respectively zero-dimensional, respectively realcompact), and (c) G is σ-compact if and only if G+ normal. We end the paper by showing the existence of a discrete group that is not [MOORE] and which still respects compactness.