Distal compactifications of group extensions
type: Item
instance of: scholarly article
title: Distal compactifications of group extensions
zbMATH Open document ID: 0928.43004
author: Hugo D. Junghenn
author: Paul Milnes
published in: Rocky Mountain Journal of Mathematics
publication date: 2000-01-09T00:00:00Z
full work available at URL: CONT29-1.html
review text: Let \(e\to N\to G\to K\to e\) be a short exact sequence of groups. By a result of Schreier, \(G\) is then isomorphic to \(N\times K\) with multiplication given by \((s,t)(s',t')= (st(s')[t,t'], tt')\), \(s,s'\in N,t,t'\in K\), where the Schreier mappings \(K\times K\to N\), \((t,t')\to [t,t']\) and \(N\times K\to N\), \((s,t)\to t(s)\) satisfy certain extension conditions. Suppose that \(N\) and \(K\) are topological groups and that the Schreier maps are continuous. Then \(G\) is a topological group. This paper is concerned with describing several right topological semigroup compactifications of \(G\) in terms of the corresponding compactifications of \(N\) and \(K\). As a sample, consider \(G^{LC}\), the universal right topological semigroup compactification with the property that the natural left action of \(G\) on the compactification is jointly continuous. The problem is whether \(G^{LC}\) is a right topological semigroup extension of some compactification of \(N\) with \(K^{LC}\). Simple examples show that such a result is unlikely unless \(K\) or \(N\) is compact. Suppose that \(N\) is compact. Then \(G^{LC}\) is the Schreier extension of \(N\) by \(K^{LC}\) if and only if the set \([\cdot,K]\) is equicontinuous. In particular, if \(N\) is compact and \(G\) is a semi-direct product of \(N\) and \(K\), then \(G^{LC}\) is a semi-direct product of \(N\) and \(K^{LC}\). A similar result is obtained when \(K\) is compact. The authors also investigate distal and point distal compactifications and obtain various results in the above spirit. Several illustrating examples are given. The almost periodic compactification is studied elsewhere.
reviewed by: Eberhard Kaniuth
Mathematics Subject Classification ID: 43A60
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 1309617
zbMATH Keywords: group extension
zbMATH Keywords: left norm continuous
zbMATH Keywords: topological groups
zbMATH Keywords: topological semigroup compactifications
zbMATH Keywords: distal and point distal compactifications
MaRDI profile type: Publication
OpenAlex ID: W2013390679
DOI: 10.1216/RMJM/1181071687
description: scientific article; zbMATH DE number 1309617