The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group G.& nbsp; A deep result due to Ernest & nbsp; [13]& nbsp; and Hughes & nbsp; [22 ] asserts that every weakly compact subset of a locally compact & nbsp; (LC)& nbsp; group G is compact in the LC-topology, generalizing thereby a previous result of Glicksberg & nbsp; [19] for abelian locally compact & nbsp; (LCA)& nbsp; 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 countable compactness, pseudocompactness and functional boundedness.& nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; (c) 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).