Straightening theorem for bounded Abelian groups
type: Item
instance of: scholarly article
title: Straightening theorem for bounded Abelian groups
zbMATH Open document ID: 1160.22002
author: Lorenzo de Leo
author: Dikran Dikranjan
published in: Topology and its Applications
publication date: 2008-11-12T00:00:00Z
review text: For any abelian group \(G\), let \(G^{\sharp}\) denote the group \(G\) equipped with its Bohr topology, i. e. the initial topology of \(G\) with respect to the family of all homomorphisms of \(G\) into the circle group. For a positive integer \(m\) and a cardinal \(\kappa,\) let \({\mathbb V}_m^{\kappa}=\bigoplus_{\kappa} {\mathbb Z_m},\) where \({\mathbb Z}_m\) denotes the cyclic group of order \(m\). The authors prove that for any bounded abelian group \(J\), any \(m>1\) and any continuous function \(\pi:({\mathbb V_m^{\mathbb N}})^{\sharp}\to J^{\sharp},\) there exists an infinite subset \(A\) of \({\mathbb N}\) and a homomorphism \(\ell: {\mathbb V_m^{\mathbb N}}\to J\) such that \(\pi\) coincides with \(\ell\) on the set of all elements of \({\mathbb V_m^A}\) with at most \(m\) nonzero coordinates. Using the Prüfer structure theorem, this result may be briefly put as follows: every continuous function between two bounded abelian groups endowed with their Bohr topologies coincides with a homomorphism when restricted to an infinite subset of the domain. This extends results by \textit{K. Kunen} [Topology Appl. 90, 97--107 (1998, Zbl 0974.54025)] and by \textit{D. Dikranjan} and \textit{S. Watson} [J. Pure Appl. Algebra 163, 147--158 (2001; Zbl 0983.22001)]. A number of applications of this ``straightening'' theorem, mainly concerning Bohr homeomorphisms, is given.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 5365231
zbMATH Keywords: Bohr topology
zbMATH Keywords: bounded abelian groups
zbMATH Keywords: Ulm-Kaplansky invariants
reviewed by: X. Domínguez
MaRDI profile type: Publication
full work available at URL: j.topol.2007.07.010
OpenAlex ID: W2043545124
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DOI: 10.1016/J.TOPOL.2007.07.010
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description: scientific article; zbMATH DE number 5365231