Arcs in the Pontryagin dual of a topological abelian group
type: Item
instance of: scholarly article
title: Arcs in the Pontryagin dual of a topological abelian group
zbMATH Open document ID: 1332.22010
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-16T00:00:00Z
review text: Let \(G\) be a topological abelian group. The dual group \(\widehat G\) of \(G\) is defined as the group of all continuous homomorphisms (characters) of \(G\) into the unit circle of the complex plane \(\mathbb S^1\) and it is equipped with the compact open topology. With this topology \(\widehat G\) is a Hausdorff topological abelian group, which is generally called Pontryagin dual of \(G\). Let \(\widehat G_a\) denote the arc-component of \(\widehat G\), \(\widehat G_{\mathrm{lift}}\) the group of all characters which can be lifted over the real numbers \(\mathbb R\), and \(\widehat G_{\mathrm{lift}\sigma}\) the group of those characters that can be lifted over \(\mathbb R\) equipped with its Bohr topology (the topology of pointwise convergence on \(\widehat {\mathbb R}\)). If \(\exp_G\) denotes the exponential function of \(G\), the elements of its image \(\mathrm{im} \exp_G\) are those lying in one-parameter subgroups, and \(G\) is the union of its one-parameter subgroups if and only if \(\exp_G\) is onto. Among others, the following results are proved: (1) in Theorem 3.6, it is established that \(G_{\mathrm{lift}}\subseteq G_{\mathrm{lift}\sigma}\subseteq \mathrm{im} \exp_{\widehat G}\subseteq \widehat G_a\). Furthermore, in analyzing conditions under which these equalities hold, the authors introduce the equicontinuous arc property (EAP for short), which is satisfied for a topological abelian group \(G\) if every arc in \(\widehat G\) is equicontinuous. In Theorem 3.13, it is proved that if \(G\) has EAP, then \(G_{\mathrm{lift}}= \widehat G_a\) holds. The paper also contains other interesting results and pertinent examples.
reviewed by: Salvador Hernández
Mathematics Subject Classification ID: 22D35
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 6387123
zbMATH Keywords: equicontinuous arc property
zbMATH Keywords: arcwise connected component
zbMATH Keywords: one-parameter subgroup
zbMATH Keywords: Pontryagin duality
author: Lydia Außenhofer
author: M. J. Chasco
author: X. Domínguez
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.12.038
OpenAlex ID: W2056131855
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DOI: 10.1016/J.JMAA.2014.12.038
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description: scientific article; zbMATH DE number 6387123