Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras
type: Item
instance of: scholarly article
title: Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras
zbMATH Open document ID: 0911.46030
author: John W.Baker
author: A. T.-M. Lau
author: J. Pym
published in: Journal of Functional Analysis
publication date: 1999-05-04T00:00:00Z
review text: Let \(A\) be a complex Banach algebra with second dual \(A^{**}\), also a Banach algebra. The topological centre \(\Lambda(A^{**})\) is the set of elements in \(A^{**}\) yielding a weak\(^*\)-continuous right multiplication. The authors begin with a review of results of \textit{M. Grosser} [``Bidualräume und Vervollständigungen von Banachmodulen'', Lect. Notes Math. 717 (1979; Zbl 0412.46005)] on the structure of \(A^{**}\) and \(A^*A\), in particular that the existence of a bounded right approximate identity in \(A\) implies that \(A^{**}\) is isomorphic in the category of Banach spaces and continuous linear maps to \((A^*A)^*\oplus (A^*A)^{\perp}\). They then give criteria for a weakly sequential complete Banach algebra \(A\) that \(\Lambda(A^{**})= A\) and that the only \(\lambda\) in \(\Lambda(A^{**})\) with \(\lambda A\subseteq A\) are themselves in \(A\). The next main result provides that under suitable hypotheses every continuous linear transformation \(T\) from \(A^*\) to \(A^*A\) with \(T(xf)= xTf\) is implemented by a right multiplication by an element of \(A\). Special cases are treated: group algebras, the Volterra algebra and \(\ell^1(S)\) for a commutative discrete semigroup \(S\).
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 1213733
zbMATH Keywords: topological centre
zbMATH Keywords: weak\(^*\)-continuous right multiplication
zbMATH Keywords: existence of a bounded right approximate identity
zbMATH Keywords: weakly sequential complete Banach algebra
zbMATH Keywords: group algebras
zbMATH Keywords: Volterra algebra
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
full work available at URL: jfan.1998.3280
OpenAlex ID: W1982989186
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DOI: 10.1006/JFAN.1998.3280
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description: scientific article; zbMATH DE number 1213733