Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups
type: Item
instance of: scholarly article
title: Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups
zbMATH Open document ID: 1401.22012
DOI: 10.1134/S1061920818030081
published in: Russian Journal of Mathematical Physics
publication date: 2018-10-24T00:00:00Z
review text: Let \(G\) be a connected locally compact group, let \(\mathcal N\) be the family of all compact normal subgroups \(N\subset G\) such that \(G/N\) is a Lie group, and let \(\pi\) be a locally bounded finite-dimensional representation of \(G\) in a normed space \(E\). Set \(FDG(\pi)=\cap_{N\in\mathcal N}\overline{\pi(N)}\). It is shown that if there is an \(N\in\mathcal N\) for which \(\pi(N)\) is abelian, then \(\pi\) is continuous on the commutator subgroup \(G'\) of \(G\) in the intrinsic Lie topology of this commutator subgroup, and that if \(\pi(N)\) is non-abelian for every \(N\in\mathcal N\) then the restriction of \(\pi\) to \(G'\) is discontinuous. It is also shown that if \(FDG(\pi|_{G'})\) is contained in a ball of radius less than \(\sqrt{3}\) centered at the identity operator \(\text{id}_E\) then \(FDG(\pi|_{G'})=\{\text{id}_E\}\), and \(\pi|_{G'}\) is continuous.
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22D12
Mathematics Subject Classification ID: 22B05
zbMATH DE Number: 6964581
zbMATH Keywords: locally compact group
zbMATH Keywords: Lie group
zbMATH Keywords: continuous representations on normed spaces
author: A. I. Shtern
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
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full work available at URL: s1061920818030081

Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups (OpenAlex ID)
OpenAlex ID: W2894207767

Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups (Wikidata QID)
Wikidata QID: Q129197012