The structure of \(\varphi\)-module amenable Banach algebras (Q1722270)

Summary: We study the concept of \(\varphi\)-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of \(\varphi\)-amenability and \(\varphi\)-module amenability of Banach algebras. As a consequence, we show that, if \(S\) is an inverse semigroup with finite set \(E\) of idempotents and \(l^1 \left(S\right)\) is a commutative Banach \(l^1 \left(E\right)\)-module, then \(l^1 \left(S\right)^{* *}\) is \(\varphi^{* *}\)-module amenable if and only if \(S\) is finite, when \(\varphi \in \mathrm{Hm}_{l^1 \left(E\right)} \left(l^1 \left(S\right)\right)\) is an epimorphism. Indeed, we have generalized a well-known result due to \textit{F. Ghahramani} et al. [Proc. Am. Math. Soc. 124, No. 5, 1489--1497 (1996; Zbl 0851.46035)].