Final group topologies, Kac-Moody groups and Pontryagin duality (Q611019)

In the paper under review the authors study final group topologies and their relations to compactness properties. To start with the authors study situations ensuring that a final group topology is locally compact and present a large variety of non-trivial examples. Another situations the authors are interested in are when a colimit or direct limit is a \(k_{\omega}\)-space or is locally \(k_{\omega}\). In particular, the authors show that a topological group is locally \(k_{\omega}\) if and only if it has an open subgroup which is a \(k_{\omega} \)-group. As an application, it is shown that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of \(k_{\omega}\)-groups). Further, it is shown that \(G^*\) is locally \(k_{\omega}\) for metrizable \(G\), while \(G^*\) is almost metrizable whenever \(G\) is locally \(k_{\omega}\). The second application concerns Pontryagin duality theory and the authors manage to strengthen several results from [\textit{S. Ardanza-Trevijano} and \textit{M. J. Chasco}, J. Pure Appl. Algebra 202, No. 1--3, 11--21 (2005; Zbl 1090.22001)]. In particular, the authors explore the relation between countable projective limits of almost metrizable abelian groups and countable direct limits of locally \(k_{\omega}\) abelian groups.