Weak*-closed invariant subspaces and ideals of semigroup algebras on foundation semigroups

Abstract

Let S be a locally compact foundation semigroup with identity and be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of In this paper, we prove that X is invariantly complemented in if and only if the left ideal of has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenable if and only if every complemented weak*-closed left translation invariant subspace of is invariantly complemented in .