Projections on Weak*-closed Subspace of Dual Banach Algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on Ma(S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A and B(A*) which commutes with translations and convolution.