Characteristics of the Mackey Topology for Abelian Topological Groups

Abstract

This chapter is inspired on the Mackey-Arens Theorem, and consists on a thorough study of its validity in the class of locally quasi-convex abelian topological groups. If \(G\) is an abelian group and \(H\) is a group of characters which separates points of \(G\), the pair \((G, H)\) is said to be a dual pair. Any group topology on \(G\) which has \(H\) as its group of continuous characters is said to be compatible with the pair \((G, H)\) or with the group duality \((G, H)\). If the starting group \(G\) is already a topological group, a natural duality is obtained taking \(H\) as the group of its continuous characters. The family of all locally quasi-convex topologies defined on an abelian group \(G\), with a fixed common character group \(H\), was studied for the first time in [9]. It is a problem not solved yet if the supremum of a family of locally quasi-convex compatible topologies on an abelian topological group \(G\) is again a compatible topology. If it is, then it is called the Mackey topology for \(G\). A locally quasi-convex group \(G\) is said to be a Mackey group whenever it carries the Mackey topology. Locally quasi-convex topologies can be characterized in terms of the families of equicontinuous subsets that they produce in the corresponding dual group. We have adopted this point of view and we have defined a grading of Mackey-type properties for abelian topological groups. We also study the stability of these properties through quotients.