# Module $(\varphi ,\psi )$-amenability of Banach algebras

## Abasalt Bodaghi

Address:

Department of Mathematics, Islamic AZAD University, Garmsar Branch, Garmsar, Iran

Current address: Laboratory of Theoretical Studies, Institute for Mathematical Research, University Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

E-mail:

abasalt@putra.upm.edu.my

abasalt_bodaghi@yahoo.com

Abstract: Let $S$ be an inverse semigroup with the set of idempotents $E$ and $S/\approx $ be an appropriate group homomorphic image of $S$. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra $\ell ^1(S)$ and the semigroup algebra $ {\ell ^{1}}(S/\approx )$ with coefficients in the same space. As a consequence, we prove that $S$ is amenable if and only if $S/\approx $ is amenable. This could be considered as the same result of Duncan and Namioka [5] with another method which asserts that the inverse semigroup $S$ is amenable if and only if the group homomorphic image $S/\sim $ is amenable, where $\sim $ is a congruence relation on $S$.

AMSclassification: primary 43A07; secondary 46H25.

Keywords: Banach modules, module derivation, module amenability, inverse semigroup.