In the current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra can not be L 1 -predual. We prove also that a nilpotent ideal in Johnson pseudo-contractible Banach algebras under the approximation property is forced to be zero. Among other things, we characterize the property F of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup S such that (E(S), ≤) is uniformly locally finite, l 1 (S) has the property F if and only if each maximal subgroup of S is amenable, where E(S) is the set of idempotents of S. Furthermore, we study the property F of some θ-Lau product structures.
