On dual valued operators on Banach álgebras

Abstract

Let U be a regular Banach algebra and let D:U→U∗ be a bounded linear operator, where U∗ is the topological dual space of U. We seek conditions under which the transpose of D becomes a bounded derivation on U∗∗. We focus our attention on the class D(U) of bounded derivations D:U→U∗ so that <a,D(a)>=0 for all a∈U. We consider this matter in the setting of Beurling algebras on the additive group of integers. We show that U is a weakly amenable Banach algebra if and only if D(U)≠{0}.