The dual space of precompact groups

TL;DR

This paper extends the understanding of the dual space of precompact groups, showing that the dual remains discrete for a broader class of groups beyond metrizable ones, specifically almost metrizable precompact groups.

Contribution

It generalizes previous results by proving the dual space is discrete for almost metrizable precompact groups, expanding the class of groups with this property.

Findings

The dual space of an almost metrizable precompact group is discrete.

Discreteness of the dual space holds for a broader class of groups.

Extends previous results from metrizable to almost metrizable groups.

Abstract

For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete. In an earlier paper we proved that $\hat G$ is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when $G$ is an almost metrizable precompact group.