Module and Hochschild cohomology of certain semigroup algebras

Abstract

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 \(H_{{\ell ^1}\left( E \right)}^1\) (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 \(H_{{\ell ^1}\left( E \right)}^2\) (l1(S), l1(GS)(2n+1)) is a Banach space.