On some module amenability-like properties of Banach algebras with applications

Abstract

In this paper, we introduce a new notion of module strong pseudo-amenability for Banach algebras. We study the relation between this new concept to other various notions at this issue, module pseudo-contractibility, module pseudo-amenability and module approximate amenability. For an inverse semigroup [Formula: see text] with the set of all idempotents [Formula: see text], we show that [Formula: see text] is module strong pseudo-amenable as an [Formula: see text]-module if and only if [Formula: see text] is amenable. For specific types of semigroups such as Brandt semigroups and bicyclic semigroups, we investigate the module strong pseudo-amenability of [Formula: see text]. We show that for every non-empty set [Formula: see text], [Formula: see text] under this new notion is forced to have a finite index as an [Formula: see text]-module, where [Formula: see text].