The weak compactification of locally compact groups

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.
arXiv admin note: text overlap with arXiv:1704.03438