Group representations and construction of minimal topological groups
type: Item
instance of: scholarly article
title: Group representations and construction of minimal topological groups
zbMATH Open document ID: 0824.22003
DOI: 10.1016/0166-8641(94)00043-3
published in: Topology and its Applications
publication date: 1995-11-20T00:00:00Z
review text: For every continuous biadditive mapping \(\omega : E \times F \to A\) with Abelian topological groups \(E\), \(F\) and \(A\), the author defines a corresponding topological group \(M(\omega)\) and proves its minimality under natural restrictions. With the use of the evaluation mapping \(G \times G^* \to \mathbb{T}\) provided by the Pontryagin-van Kampen duality and the canonical duality \(E \times E^* \to \mathbb{R}\) for a normed space \(E\), it is shown that every locally compact Abelian group is a group retract of a minimal locally compact group. Furthermore, every Abelian topological group is a quotient of a perfectly minimal group. The latter gives a positive answer to a question of Arkhangel'skij for the abelian case. The author also solves a problem of Stoyanov by constructing a perfectly minimal group of countable weight which is not topologically isomorphic to any subgroup of the unitary group \(U(\mathbb{H})\) for a Hilbert space \(\mathbb{H}\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 735667
zbMATH Keywords: bilinear forms
zbMATH Keywords: unconditionally closed sets
zbMATH Keywords: Abelian topological groups
zbMATH Keywords: Pontryagin-van Kampen duality
zbMATH Keywords: locally compact Abelian group
zbMATH Keywords: minimal locally compact group
zbMATH Keywords: perfectly minimal group
zbMATH Keywords: countable weight
author: Michael Megrelishvili
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)00043-3
OpenAlex ID: W1968064437
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description: scientific article; zbMATH DE number 735667