The structure of \(\varphi\)-module amenable Banach algebras. (English) Zbl 1470.46075

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 F. Ghahramani et al. [Proc. Am. Math. Soc. 124, No. 5, 1489–1497 (1996; Zbl 0851.46035)].


46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
43A07 Means on groups, semigroups, etc.; amenable groups


Zbl 0851.46035
Full Text: DOI


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