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 H ^k_A (A,X*) and H ^k(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(G _S )⁽²ⁿ⁺¹⁾) is trivial for each n ∈ N, where G _S is the maximal group homomorphic image of S. Also, the second module cohomology $H_{{\ell ^1}\left( E \right)}^2$ (ℓ¹(S), ℓ¹(G _S )⁽²ⁿ⁺¹⁾) is a Banach space.