On locally \(M\)-convex \(\ast\)-algebra bundles
type: Item
instance of: scholarly article
title: On locally \(M\)-convex \(\ast\)-algebra bundles
zbMATH Open document ID: 0942.46045
DOI: 10.1007/BF02169284
author: M. Fragoulopoulou
author: Athanasios Kyriazis
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-01-25T00:00:00Z
review text: The notion of topological algebra bundle has already been given by \textit{A. Kyriazis} in the paper [``On topological algebra bundles'', J. Aust. Math. Soc., Ser. A 43, 398-419 (1987; Zbl 0672.46045)]. Briefly, a topological algebra bundle over a Hausdorff topological space \(X\) is a triple \(\xi= (E,\pi, X)\) with \(E\) a nonempty topological space and \(\pi: E\to X\) a continuous map whose fibre \(E_x= \pi^{-1}(x)\), \(x\in X\), is a topological algebra in such a way that the standard local triviality conditions are satisfied. In particular, a topological \(*\)-algebra bundle is a topological algebra bundle whose fibres are topological \(*\)-algebras. For \(\xi\) a locally \(m\)-convex (lmc) \(*\)-algebra bundle, the authors define here the enveloping locally \(C^*\)-algebra bundle \({\mathcal E}(\xi)\) where the fibres are the enveloping locally \(C^*\)-algebra. So, the enveloping functor \({\mathcal E}\) is obtained (Theorem 3.1). Moreover, related to the sections functor \(\Gamma\), this functor satisfies (under certain conditions) \({\mathcal E}(\Gamma(\xi))= \Gamma({\mathcal E}(\xi))\) with respect to an isomorphism of locally \(C^*\)-algebras. In the last section, as an application of the functor \({\mathcal E}\) an analogue of \textit{A. Mallios'} result [``Topological algebras, selected topics'', North-Holland, Amsterdam (1986; Zbl 0597.46046), 6.4, p. 498] is obtained in the case of an lmc \(C^\infty\)-\(*\)-algebra bundle \(\xi\).
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46K99
zbMATH DE Number: 1393013
zbMATH Keywords: locally \(M\)-convex \(*\)-algebra bundles
zbMATH Keywords: topological algebra bundle
zbMATH Keywords: topological \(*\)-algebra bundle
zbMATH Keywords: enveloping locally \(C^*\)-algebra
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
cites work: Q3799170
cites work: Q4284879
cites work: Q4379868
cites work: LOCALLY C*-ALGEBRA
cites work: Q3825564
cites work: Q3730448
cites work: Vector bundles and K-theory over topological algebras
cites work: Locally multiplicatively-convex topological algebras
full work available at URL: https://doi.org/10.1007/bf02169284
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description: scientific article; zbMATH DE number 1393013