On amenable-like properties of some Banach algebras with applications to study their nilpotent ideals

Abstract

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be [Formula: see text]-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property [Formula: see text] of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup [Formula: see text] such that [Formula: see text] is uniformly locally finite, [Formula: see text] has the property [Formula: see text] if and only if each maximal subgroup of [Formula: see text] is amenable, where [Formula: see text] is the set of idempotents of [Formula: see text]. Furthermore, we study the property [Formula: see text] of some [Formula: see text]-Lau product structures.