## Module amenability and weak module amenability for second dual of Banach algebras.(English)Zbl 1266.46038

Summary: We define the weak module amenability of a Banach algebra $$\mathcal A$$ which is a Banach module over another Banach algebra $$\mathfrak A$$ with compatible actions, and show that, under some mild conditions, weak module amenability of $$\mathcal A^{**}$$ implies weak module amenability of $$\mathcal A$$. Also, among other results, we investigate the relation between module Arens regularity of a Banach algebra and module amenability of its second dual. As a consequence, we prove that $$\ell^1(S)$$ is always weakly amenable (as an $$\ell^1(E)$$-module), where $$S$$ is an inverse semigroup with an upward directed set of idempotents $$E$$.

### MSC:

 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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