Local topological structure in the LUC-compactification of a locally compact group and its relationship with Veech's theorem
type: Item
instance of: scholarly article
title: Local topological structure in the LUC-compactification of a locally compact group and its relationship with Veech's theorem
zbMATH Open document ID: 1113.22004
author: T. Budak
author: J. Pym
published in: Semigroup Forum
publication date: 2007-01-26T00:00:00Z
review text: If \(G\) is a topological group, \(G^{\text{LUC}}\) stands for the largest semigroup compactification for which the multiplication is jointly continuous. It can be obtained as the structure space of the \(C^\ast\)-algebra of all bounded left uniformly continuous functions on \(G\) (thence the term LUC in its name). In this paper a realization of \(G^{\text{LUC}}\) as a quotient of the semigroup \(\beta G_d\) (the Stone-Čech-compactification of \(G\) as a discrete space) is obtained. This quotient is the result of identifying two ultrafilters \(p,q\in \beta G_d\) whenever \(UP\cap UQ\neq \emptyset\) for every neighbourhood \(U\) of the identity of \(G\) and every \(P\in p\) and \(Q\in q\). This characterization of \(G^{\text{LUC}}\) is used to illustrate the close connection between Veech's theorem (one of the main results in the theory of LUC-compactifications stating that the map \(g\mapsto gs\) is injective on \(G\) for each \(s\in G^{\text{LUC}}\)) and local structure theorems that describe topologically neighbourhoods of points in \(G^{\text{LUC}}\). A local structure theorem was already used in \textit{J. Pym} [Semigroup Forum 59, No.~2, 171--174 (1999; Zbl 0980.22009)] to prove Veech's theorem. In this paper that local structure is improved to show that for every compact \(K\subset G\) and \(x\in G^{\text{LUC}}\) there is a neighbourhood \(V\) of \(K\) and a uniformly discrete \(P\subset G\) with \(x\in \overline{P}\) such that the multiplication map establishes a homeomorphism of \(V\times \beta P\) onto \(V\overline{P}\). Veech's theorem is an obvious direct consequence of the above local structure theorem for compact sets consisting of just two points. The latter can also be obtained as a direct consequence of Veech's theorem, suggesting a strong relationship between these two facts. Some extensions of the local structure theorem and of Veech's theorem are derived in the final part of the paper. The first makes use of the results of \textit{S. Ferri} and \t…
reviewed by: Jorge Galindo
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 54H99
zbMATH DE Number: 5118909
zbMATH Keywords: semigroup compactification
zbMATH Keywords: LUC-compactification
zbMATH Keywords: local structure theorems
zbMATH Keywords: Veech's theorem
MaRDI profile type: Publication
full work available at URL: s00233-006-0623-4
OpenAlex ID: W2049976712
DOI: 10.1007/S00233-006-0623-4
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description: scientific article; zbMATH DE number 5118909