Binz-Butzmann duality versus Pontryagin duality

Abstract

A convergence structure Ξ on a set G is a set of pairs (F,x) consisting of a filter F on X and an element x∈X satisfying a few simple axioms expressing the idea that F converges to x. A set with a convergence structure is a convergence space. It should be clear what continuous functions between convergence spaces are. A subset of a convergence space is compact if every ultrafilter on it converges to an element in it. If G is a group, then (G,Ξ) is called a convergence group if (x,y)↦xy−1:G×G→G is continuous. For an abelian topological group G let ΓG denote the group of continuous characters G→R/Z. Then ΓG is a convergence group with respect to a convergence structure for which a filter F converges to χ if for any filter H on G converging to g the filter basis F(H) converges to χ(g). The convergence group arising in this fashion is denoted by Γc (G). The topological character group endowed with the compact-open topology is written Gˆ. The authors establish the following theorem: Let G be a topological abelian group such that the canonical evaluation morphism G→Gˆˆ is continuous. Then ΓcG is locally compact (in the sense that every convergent filter contains a compact member), and the bidual Gˆˆ may be identified with a topological subgroup of ΓcΓc (G). Examples show that equality does not prevail in general as it does when G is locally compact. The examples are mostly taken from topological vector spaces. For a countable family of locally compact abelian groups Gn one knows from Kaplan's theorem that the character group of ∑Gn with the box topology is ∏Gnˆ. Then the continuous convergence structure on ∏Gnˆ is finer than the convergence structure of the product topology and coarser than the convergence structure of the box topology—in general properly so in both instances. It is shown that ∑Gn≅ΓcΓc (∑Gn) canonically and, similarly, for ∏Gn.