Banach Journal of Mathematical Analysis

A generalization of the weak amenability of Banach algebras

A. Bodaghi, M. Eshaghi Gordji, and A. R. Medghalchi

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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 $A$.

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Banach J. Math. Anal., Volume 3, Number 1 (2009), 131-142.

First available in Project Euclid: 21 April 2009

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Zentralblatt MATH identifier

Primary: 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Banach algebra homomorphism derivation $(\varphi,\psi)$-derivation weak amenability second dual


Bodaghi, A.; Eshaghi Gordji, M.; Medghalchi, A. R. A generalization of the weak amenability of Banach algebras. Banach J. Math. Anal. 3 (2009), no. 1, 131--142. doi:10.15352/bjma/1240336430.

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