Module and Hochschild cohomology of certain semigroup algebras

TL;DR

This paper explores the relationship between module and Hochschild cohomology of certain Banach algebras, providing explicit calculations for inverse semigroup algebras and establishing conditions for trivial and Banach space cohomologies.

Contribution

It establishes an isomorphism between module and Hochschild cohomology groups for commutative Banach bimodules and computes cohomologies for inverse semigroup algebras with specific coefficients.

Findings

Isomorphism between module and Hochschild cohomology spaces for certain Banach algebras.

First module cohomology is trivial for inverse semigroup algebras with specified coefficients.

Second module cohomology forms a Banach space under certain conditions.

Abstract

We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, 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 (\frac{\mathcal{A}}{J}, X^*) $ are isomorphic, where $ J $ is the closed ideal of $ \mathcal{A} $ generated by the elements of the form $ a (\alpha \cdot b)-(a\cdot \alpha)b$ with $ a,b \in \mathcal{A} $ and $\alpha \in \mathfrak{A}. $ As an example, we calculate the module cohomologies of inverse semigroup algebras with coefficients in some related function algebras. In particular, we show that for an inverse semigroup $ S $ with the set of idempotents $ E $, when $\ell^1(E) $ acts on $\ell^1(S) $ by multiplication from right and trivially from 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} $. As a consequence we conclude that the second module cohomology $\mathcal{H}^2_{\ell^1(E)} (\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is a Banach space, where $ G_S $ is the maximal group homomorphic image of $ S $.