On dual-valued operators on Banach algebras

摘要

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