(Super) module amenability, module topological centre and semigroup algebras

Abstract

For a Banach algebra $\mathcal{A}$ which is also an $\mathfrak{A}$-bimodule, we study relations between module amenability of closed subalgebras of $\mathcal{A}''$, which contains $\mathcal{A}$, and module Arens regularity of $\mathcal{A}$ and the role of the module topological centre in module amenability of $\mathcal{A}''$. Then we apply these results to $\mathcal{A}=l^{1}(S)$ and $\mathfrak{A}=l^{1}(E)$ for an inverse semigroup S with subsemigroup E of idempotents. We also show that l ¹(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group $\frac{S}{\approx}$, is amenable. Moreover, we define super module amenability and show that l ¹(S) is super module amenable if and only if $\frac{S}{\approx}$ is finite.