The weak compactification of locally compact groups

TL;DR

This paper explores the properties of the weak topology generated by irreducible unitary representations of locally compact groups, focusing on how certain compact-like properties are preserved when transitioning to the original topology.

Contribution

It surveys recent results and introduces new findings on the preservation of compact-like properties from the weak topology to the original topology in locally compact groups.

Findings

Weakly compact subsets in LC groups are compact in the LC-topology.

Certain compact-like properties are preserved from the weak topology to the original topology.

Abstract

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countably compactness, pseudocompactness and functional boundedness.