Continuous and Pontryagin duality of topological groups

TL;DR

This paper compares two extensions of Pontryagin duality for topological groups beyond the locally compact case, analyzing their properties, differences, and categorical aspects.

Contribution

It extends the comparison of Pontryagin and continuous dualities, highlighting their properties and categorical differences for non-locally compact topological groups.

Findings

Pontryagin dual retains topological group structure

Continuous dual often lacks topological group structure

Continuous dual has favorable categorical properties

Abstract

For Pontryagin's group duality in the setting of locally compact topological Abelian groups, the topology on the character group is the compact open topology. There exist at present two extensions of this theory to topological groups which are not necessarily locally compact. The first, called the Pontryagin dual, retains the compact-open topology. The second, the continuous dual, uses the continuous convergence structure. Both coincide on locally compact topological groups but differ dramatically otherwise. The Pontryagin dual is a topological group while the continuous dual is usually not. On the other hand, the continuous dual is a left adjoint and enjoys many categorical properties which fail for the Pontryagin dual. An examination and comparison of these dualities was initiated in \cite{CMP1}. In this paper we extend this comparison considerably.