Arens semi-regular Banach algebras
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.
scientific article; zbMATH DE number 3870013
scientific article