A GENERALIZATION OF THE WEAK AMENABILITY OF BANACH ALGEBRAS

Let A be a Banach algebra and let phi and psi be continuous homomorphisms on A. We consider the following module actions on A, a.x = phi(a)x, x.a = x psi(a) (a, x is an element of A). We denote by A((phi,psi)) the above A-module. We call the Banach algebra A, (phi,psi)-weakly amenable if every derivation from A into (A((phi,psi)))* is inner. In this paper among many other things we investigate the relations between weak amenability and (phi,psi)-weak amenability of A. Some conditions can be imposed on A such that the (phi '',psi '')-weak amenability of A** implies the (phi,psi)-weak amenability of A.