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 $\mathcal{A}$-$\mathfrak{A}$-bimodule $X$ and every $k\in\mathbb{N}$, the seminormed spaces $\mathcal{H}^{k}_{\mathfrak{A}}(\mathcal{A},X^*)$ and $\mathcal{H}^k(\mathcal{A}/J,X^*)$ are isomorphic, where $J$ is a specific closed ideal of $\mathcal{A}$. As an example, we show that, for an inverse semigroup $S$ with the set of idempotents $E$, where $\ell^1(E)$ acts on $\ell^1(S)$ by multiplication on the right and trivially on the left, the first module cohomology $\mathcal{H}^1_{\ell^1(E)}(\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is trivial for each $n\in\mathbb{N}$, where $G_S$ is the maximal group homomorphic image of $S$. Also, the second module cohomology $\mathcal{H}^2_{\ell^1(E)}(\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is a Banach space.