Group duality with the topology of precompact convergence

Abstract

The Pontryagin–van Kampen (P–vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P–vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P–vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.