Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras
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 \(S\).
weak\(^*\)-continuous right multiplication
existence of a bounded right approximate identity
weakly sequential complete Banach algebra
scientific article; zbMATH DE number 1213733
scientific article