Module and Hochschild cohomology of certain semigroup algebras

We study the relation between the module and Hochschild cohomology groups of Banach algebras.We show that, for every commutative Banach A-A-bimodule X and every k ∈ N, the seminormed spaces HAk (A,X*) and Hk(A /J,X*) are isomorphic, where J is a specific closed ideal of A. As an example, we show that, for an inverse semigroup S with the set of idempotents E, where ℓ1(E) acts on ℓ1(S) by multiplication on the right and trivially on the left, the first module cohomology \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{\ell ^1}\left( E \right)}^1$$\end{document} (ℓ1(S), ℓ1(GS)(2n+1)) is trivial for each n ∈ N, where GS is the maximal group homomorphic image of S. Also, the second module cohomology \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{\ell ^1}\left( E \right)}^2$$\end{document} (ℓ1(S), ℓ1(GS)(2n+1)) is a Banach space.