The weak compactification of locally compact groups

DOI10.1016/J.TOPOL.2021.107917arXiv2102.12207MaRDI QIDQ6361436

Publication date: 24 February 2021

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.

Mathematics Subject Classification ID

Topological groups (topological aspects) (54H11) Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) (43A46) Unitary representations of locally compact groups (22D10) General properties and structure of locally compact groups (22D05) Duality theorems for locally compact groups (22D35) Character groups and dual objects (43A40)

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