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 \(E\).


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