On locally \(A\)-convex modules
type: Item
instance of: scholarly article
title: On locally \(A\)-convex modules
zbMATH Open document ID: 1483.46050
author: Marina Haralampidou
author: Lourdes Palacios F.
author: Carlos Signoret
author: M. Oudadess
published in: Mediterranean Journal of Mathematics
publication date: 2022-01-17T00:00:00Z
review text: The paper deals with the inheritance of the property of being some type of an \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). More precisely, the authors are interested in the following classes of \(\mathcal A\)-modules:\par Let \((E, (\mid\!\cdot\!\mid_\lambda)_{\lambda\in\Lambda})\) be a left \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). Then \(E\) is\par a) a left \(\mathcal A\)-module with a jointly continuous action if, for each \(\lambda\in \Lambda\), there exist \(i(\lambda)\in I\) and \(\lambda'\in\Lambda\) such that \(\mid ax\!\mid_\lambda\leqslant\mid \!a\!\mid_{i(\lambda)}\mid\!x\mid_{\lambda'}\) for every \(a\in\mathcal A\) and \(x\in E\);\par b) a locally \(A\)-convex \(\mathcal A\)-module if, for each \(a\in\mathcal A\) and each \(\lambda\in\Lambda\), there exists \(\alpha(a, \lambda)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a, \lambda)\mid\! x\mid_\lambda\) for every \(x\in E\);\par c) a locally uniformly \(A\)-convex left \(\mathcal A\)-module if, for each \(a\in\mathcal A\), there exists \(\alpha(a)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a)\mid\! x\mid_\lambda\) for every \(\lambda\in\Lambda\) and \(x\in E\);\par d) a locally \(m\)-convex left \(\mathcal A\)-module if, for each \(\lambda\in\Lambda\), there exists \(i(\lambda)\in I\) such that \(\mid ax\!\mid_\lambda\leqslant\mid\!a\!\mid_{i(\lambda)}\mid\! x\mid_\lambda\) for every \(a\in\mathcal A\) and \(x\in E\).\par Many examples of algebras, belonging to the one of the classes and not in the another, are provided.\par The authors show that the property of being locally \(A\)-convex left \(\mathcal A\)-module is inherited by the operations of\par (i) taking the quotient by a left \(\mathcal A\)-submodule;\par (ii) considering instead of a non-unital algebra \(\mathcal A\) its unitization;\par (iii) taking a completion of a locally \(A\)-convex left \(\mathcal …
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 13Jxx
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 46M40
zbMATH DE Number: 7458064
zbMATH Keywords: locally \(m\)-convex algebra
zbMATH Keywords: locally \(A\)-convex algebra
zbMATH Keywords: locally uniformly \(A\)-convex algebra
zbMATH Keywords: left \(\mathcal{A}\)-module
zbMATH Keywords: locally \(A\)-convex left module
zbMATH Keywords: locally uniformly \(A\)-convex left module, Arens-Michael-like decomposition
reviewed by: Mart Abel
MaRDI profile type: Publication
full work available at URL: s00009-021-01935-x
OpenAlex ID: W4206231262
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DOI: 10.1007/S00009-021-01935-X
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description: scientific article; zbMATH DE number 7458064