A characterization of the maximally almost periodic abelian groups
type: Item
instance of: scholarly article
title: A characterization of the maximally almost periodic abelian groups
zbMATH Open document ID: 1065.22003
author: Dikran Dikranjan
author: Chiara Milan
author: Alberto Tonolo
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-18T00:00:00Z
review text: For an abelian topological group \(G\), let \(\hat G\) denote its Pontryagin dual group (i.e., the group of all continuous homomorphisms of \(G\) into the circle group, \(\mathbb T\), with pointwise multiplication as composition law). Given a sequence \(\underline{u}=(u_n)\in \hat G^{\mathbb N}\), consider the subgroup \(s_{\underline{u}}(G)=\{x\in G:u_n(x)\rightarrow 0\;\text{in}\;G\}\). If \(H\) is a subgroup of \(G\), the authors define its Galois-closure (\(\mathfrak g\)-closure) by \[ \mathfrak g_G(H)=\bigcap \{s_{\underline{u}}(G):\underline{u}\in \hat G^{\mathbb N}, H\leq s_{\underline{u}}(G) \}. \] Thus, a subgroup \(H\) is \(\mathfrak g\)-{closed} if \(\mathfrak g_G(H)=H\) and \(\mathfrak g\)-\textit{dense} if \(\mathfrak g_G(H)=G\). Finally, it is said that a topological abelian group \(G\) has the \(\mathfrak g\)-{closure property} if every cyclic subgroup of \(G\) is \(\mathfrak g\)-closed. The main result of this paper is the following. \textbf{Theorem:} A topological abelian group is MAP if and only if \(G\) has the \(\mathfrak g\)-closure property. This extends a previous result which deals with the circle group and, thereby, affirms an appropriate version of a conjecture given by \textit{A. Bíró}, \textit{J.-M. Deshouillers} and \textit{V. T. Sós} in [Studia Sci. Math. Hungar. 38, 97--113 (2001; Zbl 1006.11038)]. The main result is established in several steps. In Section 2 the authors point out the basic properties of the Galois closure \(\mathfrak g_G(-)\) and reduce the problem to precompact groups. In Section 3 they analyze the structure of the subgroups \(s_{c^*}(\mathbb Z_p)\), when \(c^*\) is a subsequence of the canonical set of generators of the Prüfer group \(Z(p^\infty)\) and \(\mathbb Z_p\) denotes the group of \(p\)-adic integers. The following result of independent interest is accomplished. \textbf{Theorem:} Let \((n_k)\) be a strictly increasing sequence of naturals and let \(\underline{c^*}=(c_{n_k})\). Then the following are equivale…
reviewed by: Salvador Hernández
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 2157518
zbMATH Keywords: maximally almost periodic abelian group
zbMATH Keywords: Galois correspondence
zbMATH Keywords: Galois closure
zbMATH Keywords: Pontryagin dual group
zbMATH Keywords: TB-sequence
zbMATH Keywords: group of p-adic integers
zbMATH Keywords: circle group
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.021
OpenAlex ID: W2145684772
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DOI: 10.1016/J.JPAA.2004.08.021
description: scientific article; zbMATH DE number 2157518