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A generalization of the weak amenability of Banach algebras. (English) Zbl 1163.46034
Summary: Let $A$ be a Banach algebra and let $\varphi$ and $\psi$ be continuous homomorphisms on $A$. We consider the following module actions on $A$, $$a\cdot x = \varphi(a)x,\quad x\cdot a = x\psi(a)\quad (a, x\in A).$$ 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 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 $A$.

46H25Normed modules and Banach modules, topological modules
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