A note on the \({\mathcal WAP}\)-compactification and the \({\mathcal LUC}\)-compactification of a topological group
type: Item
instance of: scholarly article
title: A note on the \({\mathcal WAP}\)-compactification and the \({\mathcal LUC}\)-compactification of a topological group
zbMATH Open document ID: 1057.22002
published in: Semigroup Forum
publication date: 2004-10-01T00:00:00Z
review text: 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 \(G\).
reviewed by: Vladimir G. Pestov
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 43A60
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 2105095
zbMATH Keywords: weakly almost periodic functions
zbMATH Keywords: WAP compactification
zbMATH Keywords: greatest ambit
zbMATH Keywords: semigroup compactification
zbMATH Keywords: SIN groups
zbMATH Keywords: IN groups
author: Stefano Ferri
author: D. Strauss
MaRDI profile type: Publication
full work available at URL: s00233-003-0026-8
OpenAlex ID: W2144131218
DOI: 10.1007/S00233-003-0026-8
Recommended article: On the actions of a locally compact group on some of its semigroup compactifications
Recommended article: A note on locally compact subsemigroups of compact groups
Recommended article: Existence and structure theorems for semigroup compactifications
Recommended article: Local topological structure in the LUC-compactification of a locally compact group and its relationship with Veech's theorem
Recommended article: A note on the WAP-compactification of groups
Recommended article: Q3759056
Recommended article: Compactifications of topological groups
Recommended article: Quasi-uniformities on topological semigroups and bicompletion
Recommended article: Q3798894
Recommended article: Distal compactifications of group extensions
description: scientific article; zbMATH DE number 2105095