A generalization of the weak amenability of Banach algebras

Abstract

Let $A$ be a Banach algebra and let $\varphi$ and $\psi$ be continuous homomorphisms on $A$. We consider the following module actions on $A$, $$a\cdot x=\varphi(a)x , \hspace{0.7cm} x\cdot a=x\psi(a) \hspace{1.5cm} (a,x\in A).$$ We denote by $A_{(\varphi,\psi)}$ the above $A$-module. We call the Banach algebra $A$, $(\varphi,\psi)$-weakly amenable if every derivation from $A$ into $(A_{(\varphi,\psi)})^*$ is inner. In this paper among many other things we investigate the relations between weak amenability and $(\varphi,\psi)$-weak amenability of $A$. Some conditions can be imposed on $A$ such that the $(\varphi'',\psi'')$-weak amenability of $A^{**}$ implies the $(\varphi,\psi)$-weak amenability of $A$.