A note on the \({\mathcal WAP}\)-compactification and the \({\mathcal LUC}\)-compactification of a topological group
The first main theorem concerns the canonical morphism, \(\pi\), of semitopological semigroups from the greatest ambit (\(LUC\)-compactification) of a topological group \(G\) (that is, the maximal equivariant compactification) onto the WAP-compactification (the compactification corresponding to the \(C^\ast\)-algebra of all weakly almost periodic functions on \(G\)). It is shown that if \(G\) is a locally compact SIN-group (a topological group whose left and right uniformities coincide), then the greatest ambit of \(G\) contains a dense open subset of points of unicity for \(\pi\). As a consequence, the cardinality of the WAP compactification of an \(IN\)-group (a group having an invariant compact neighbourhood of identity) is estimated from below. As another consequence, the action of a locally compact SIN group \(G\) on the WAP-compactification is free on a dense open subset of points. The second part deals with the one-sided (Weil) completions of a topological group. While some of the results from this part are known [see e.g. \textit{W. Roelcke} and \textit{S. Dierolf}, Uniform structures on topological groups and their quotients (New York etc. 1981; Zbl 0489.22001)], there are some new results as well, such as a characterization of the Weil completion of a topological group in terms of points of joint continuity in the greatest ambit of \(G\).
scientific article; zbMATH DE number 2105095