The Equality of Weak Amenability and Weak Module Amanability for Semigroup Algebra of Commutative Inverse Semigroups

Abstract

‎Let $S$ be a commutative inverse semigroup with idempotent set $E$‎. ‎In this case $\ell^1(S)$ is a commutative Banach $\ell^1(E)$-module with actions‎ ‎$\delta_s\cdot \delta_e=\delta_e\cdot\delta_s=\delta_{se},$ where $\delta_e$ and $\delta_s$ are the point masses at $e\in E$ and $s\in S$‎, ‎respectively‎. ‎In this paper‎, ‎we will show that‎ ‎\begin{equation*}‎ ‎\HH^1(\ell^1(S)‎, ‎{\ell^1(S)}^{(2n-1)})\simeq \HH^1_{\ell^1(E)}(\ell^1(S)‎, ‎{\ell^1(S)}^{(2n-1)})\qquad\qquad (n\in \mathbb{N})‎. ‎\end{equation*}‎ ‎But it is well known that the semigroup algebra $\ell^1(S)$ is always $(2n-1)$-weakly amenable (by \cite[Theorem 2.1]{BD} and \cite[Theorem 2.8.63]{D})‎. ‎So our results show that $\ell^1(S)$ is always weakly module amenable (as a $\ell^1(E)$-module)‎, ‎which confirms the correctness of Theorem 2.1 of \cite{NP} and Theorem 4.1 of \cite{AB}‎, ‎but with a much easier path‎.