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

Let S be a locally compact foundation semigroup with identity and M_a (S) be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of ?M_(a ) (S)?^*. In this paper, we prove that X is invariantly complemented in ?M_(a ) (S)?^* if and only if the left ideal X_? of M_a (S) 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 L^? (S,M_a (S)) is invariantly complemented in L^? (S,M_a (S)).