On Mackey topology for groups

What is it about?

The Mackey-Arens Theorem is a wellknown result in the theory of locally convex spaces (lcs). A counterpart of Mackey-Arens Theorem is given in this paper for some classes of abelian topological groups. The seminal notion of Vilenkin, of quasi-convex subsets of an abelian topological group, is the cornerstone to permit extensions of results of lcs to abelian locally quasi-convex (lqc) topological groups. In particular, it follows from our results that the locally compact abelian groups are Mackey groups. In other words, if (G,t) is a locally compact abelian group, then t is the finest topology on G among all those locally quasi-convex topologies on G that produce the same character group.

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Why is it important?

The Mackey topology for a group was defined through the results of our paper. After its publication several authors worked on it and produced a vast literature on the topic. Some of the questions left open were solved 20 years later.