M.J. Chasco

Topologies on the direct sum of topological Abelian groups

We prove that the asterisk topologies on the direct sum of topological Abelian groups, used by Kaplan and Banaszczyk in duality theory, are different. However, in the category of locally quasiconvex groups they do not differ, and coincide with the coproduct topology.  2003 Elsevier B.V. All rights reserved. MSC: 22A05

Completeness properties of locally quasi-convex groups

It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness

An approach to duality on abelian precompact groups

Abstract We prove that every dense subgroup of a topological abelian group has the same ‘convergence dual’ as the whole group. By the ‘convergence dual’ we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does.