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)).