The Bohr topology of Moore groups
Let \(G\) be a locally compact group. Two other topologies may be considered on \(G\). First, the Bohr topology, i.e. the topology induced on \(G\) by its Bohr compactification, which is also, by a result of John von Neumann, the weakest topology for which the finite-dimensional representations of \(G\) are still continuous. Equipped with this topology, \(G\) will be denoted by \(G^+\). \(G\) can also be endowed with the weak topology, which is the weakest topology on \(G\) for which all the irreducible unitary representations of \(G\) remain continuous. With this topology, \(G\) will be written \(G^w\). The Bohr topology is coarser than the weak topology, which in turn is coarser than the original topology. These three topologies coincide when \(G\) is compact, and the first two ones do when \(G\) is Abelian. The paper under review is devoted to extend, in various directions, Glicksberg's result saying that, for Abelian locally compact groups, the compact sets are the same for \(G^+\) and for \(G\). The authors begin by characterizing groups \(G\) such that \(G^+=G^w\) as those for which all the irreducible unitary representations are finite-dimensional, that is Moore groups. Saying that \(G\) respects a topological property \({\mathcal P}\) whenever a subset \(A\) of \(G\) has \({\mathcal P}\) if and only if it has \({\mathcal P}\) in \(G^+\), it is shown that Moore groups respect compactness, pseudo-compactness, relatively functional boundedness, and to be Lindelöf. It is also shown that a Moore group \(G\) is connected (resp. zero-dimensional, resp. realcompact) if and only if \(G^+\) has the same property, and that \(G\) is \(\sigma\)-compact if and only if \(G^+\) is normal. A locally compact group \(G\) is said MAP (maximally almost periodic) when \(G^+\) is Hausdorff. Since, by the Gel'fand-Raĭkov theorem, \(G^w\) is always Hausdorff, Moore groups are MAP. The authors end their paper by giving examples of: a discrete MAP group which does not respect compactness, a discrete MAP group which respects compactness, though not being a Moore group, and a MAP group for which \(G\), \(G^w\), and \(G^+\) are pairwise different.
scientific article; zbMATH DE number 1367518