Duality theory for convergence groups

Abstract

The well-known Pontryagin Duality Theorem states that a locally compact, commutative topological group is isomorphic to its second character group, i.e., the character group of its character group. Here the character group carries the compact-open topology. There are various papers dealing with generalizations of the theorem to not necessarily locally compact commutative groups. In 1971 we suggested to replace the compact-open topology in the general case by the continuous convergence structure. This structure coincides with the compact-open topology if the group is locally compact and gives at least better categorical properties in the so-called c-duality theory. We just mention the fact that the natural mapping from a convergence group (always assumed to be commutative) into its second c-character group is always continuous. In recent years this approach has attracted again attention. In this paper we study the behaviour of the c-duality under the usual topological constructions. We show that the c-character group of a product of convergence groups is isomorphic the the coproduct (in the category of convergence groups) of the c-character groups and that the c-character group of a coproduct is isomorphic the the product of the c-character groups. So products and coproducts of c-reflexive convergence groups are c-reflexive again. Also the character group of a quotient group is isomorphic to its annihilator group. As it is well known, subgroups are very difficult to handle, since its characters can not always be extended to the whole group. We give sufficient conditions to guarantee that the c-character group of a subgroup is isomorphic to a quotient of the c-character group of the whole group and also that a subgroup of a c-reflexive group is c-reflexive. In the last section we handle topological groups. We show that the c-character group of a topological group is locally compact and so the second c-character group is also topological. We show that the natural mapping from the group into its second character group is an embedding if and only if the group is locally quasi-convex, a notion introduced by Banaszczyk in 1991.