Direct sums of spaces of functions on semigroups
type: Item
instance of: scholarly article
title: Direct sums of spaces of functions on semigroups
zbMATH Open document ID: 0807.22004
DOI: 10.1007/BF02573476
published in: Semigroup Forum
publication date: 1995-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/135339
review text: Let \(F\) be an algebra of continuous bounded functions on a semigroup \(S\) which has a separately continuous multiplication. Denote by \(S^ F\) the carrier space of \(F\), so that \(S^ F\) contains an image \(\varepsilon(S)\) of \(S\); \(S^ F\) is compact and has a natural semigroup multiplication for which right translations are continuous if \(F\) satisfies the condition known as \(m\)-admissibility. If \(G\) is an \(m\)-admissible subalgebra of \(F\), there is a continuous surjective homomorphism \(\theta : S^ F \to S^ G\). The main theorem of the paper is that, in this situation, \(F\) is a direct sum \(G \oplus H\) where \(H\) is an ideal in \(F\) invariant under left translation by elements of \(S\) if and only if there is a continuous homomorphism \(\widetilde{\theta} : S^ G \to S^ F\) such that \(\widetilde{\theta}(S^ G)\) is a left ideal in \(S^ F\) and \(\theta \circ \widetilde{\theta}\) is the identity on \(S^ G\); also \(H\) is \(m\)-admissible if and only if \(\widetilde{\theta} (S^ G)\) is an ideal. Several general consequences are described, including de Leeuw and Glicksberg's classical result about the decomposition of weakly almost periodic functions. The author also applies his theorem to give a generalization of the Furstenberg-Namioka structure theorem for compact right topological groups. This applies to compact right topological semigroups which have dense topological centres and unique minimal right ideals. The general idea is that such a semigroup can be expressed as a projective limit of a well ordered family of quotient semigroups, in which between each semigroup and its successor a joint continuity property holds, but for the (complicated) details the paper should be consulted.
Mathematics Subject Classification ID: 22A20
Mathematics Subject Classification ID: 43A60
zbMATH DE Number: 611120
zbMATH Keywords: algebra of continuous bounded functions
zbMATH Keywords: continuous multiplication
zbMATH Keywords: right translations
zbMATH Keywords: left ideal
zbMATH Keywords: weakly almost periodic functions
zbMATH Keywords: compact right topological groups
zbMATH Keywords: projective limit
zbMATH Keywords: quotient semigroups
zbMATH Keywords: joint continuity
zbMATH Keywords: \(m\)-admissible subalgebra
author: Hugo D. Junghenn
reviewed by: J. Pym
MaRDI profile type: Publication
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OpenAlex ID: W1986578984
description: scientific article; zbMATH DE number 611120