A survey on reflexivity of abelian topological groups

Abstract

The Pontryagin duality theorem for locally compact abelian groups (briefly, LCA groups) has been the starting point for many different routes of research in Mathematics. From its appearance there was a big interest to extend it in a context broader than LCA groups. Kaplan in the 40ʼs proposed—and it still remains open—the problem of characterization of all abelian topological groups for which the canonical mapping into its bidual is a topological isomorphism, assuming that the dual and the bidual carry the compact-open topology. Such groups are called reflexive.

In this survey we deal with results on reflexivity of certain classes of groups, with special emphasis on the smaller class which better reflects the properties of LCA groups, namely that of strongly reflexive groups. A topological abelian group is said to be strongly reflexive if all its closed subgroups and its Hausdorff quotients as well as the closed subgroups and the Hausdorff quotients of its dual group are reflexive.