Dual topologies on non-abelian groups

TL;DR

This paper extends the concept of locally quasi-convex groups to non-abelian groups using bornologies on representations, establishing dual topologies and relating structural properties to topological features, with applications to metrizability.

Contribution

It introduces a framework for dual topologies on non-abelian groups via bornologies on their representations, generalizing abelian results and linking structural and topological properties.

Findings

Lattice of Hausdorff totally bounded topologies is isomorphic to certain subsets of representations.

Structural properties of dual groups relate to topological properties of bornologies.

If all dense subgroups have uniformly isomorphic duals, then the group is metrizable.

Abstract

The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set $\hbox{rep}(G)$ of all finite dimensional continuous representations on a topological group $G$ in order to associate well behaved group topologies (dual topologies) to them. As a consequence, the lattice of all Hausdorff totally bounded group topologies on a group $G$ is shown to be isomorphic to the lattice of certain special subsets of $\hbox{rep}(G_d)$. Moreover, generalizing some ideas of Namioka, we relate the structural properties of the dual topological groups to topological properties of the bounded subsets belonging to the associate bornology. In like manner, certain type of bornologies that can be defined on a group $G$ allow one to define canonically associate uniformities on the dual object $\hat G$. As an application, we prove that if for every dense subgroup $H$ of a compact group $G$ we have that $\hat H$ and $\hat G$ are uniformly isomorphic, then $G$ is metrizable. Thereby, we extend to non-abelian groups some results previously considered for abelian topological groups.