Arens semi-regular Banach algebras
type: Item
instance of: scholarly article
title: Arens semi-regular Banach algebras
zbMATH Open document ID: 0546.46039
DOI: 10.1007/BF01536907
author: Michael Grosser
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178189
review text: Some important Banach algebras are not Arens regular, i.e. the two canonical extensions of the multiplication given on the algebra to its bidual are different. Among the non-regular algebras (possessing a two- sided bounded approximate identity) there are some for which these so- called Arens products agree at least on certain (large) subspaces of the respective bidual. It turns out that the class of these Banach algebras (which we call Arens semi-regular) can be characterized resp. defined in terms of a certain embedding property of the space of double multipliers of the algebra with respect to its bidual. It is shown that all commutative and all regular Banach algebras are semi-regular; the algebra of compact operators on a Banach space X is semi-regular if the dual of X has the Radon-Nikodym property. \textit{V. Losert} and \textit{H. Rindler} have shown recently [Lect. Notes Math. 1064, 368-378 (1983)] that a group algebra \(L^ 1(G)\) is semiregular if and only if G is discrete or abelian. Therefore it seems reasonable to expect that also in other cases, semi-regularity of ''classical'' Banach algebras might correspond to certain ''classical'' properties of the objects being considered.
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 47L10
zbMATH DE Number: 3870013
zbMATH Keywords: Arens regular
zbMATH Keywords: Arens products
zbMATH Keywords: Arens semi-regular
zbMATH Keywords: space of double multipliers
zbMATH Keywords: Radon-Nikodym property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3870013