Projections on weak*-closed subspaces

Let G be a locally compact group and S a weak*-closed translation invariant subspace of L∞ (G). M.E.B. Bekka proved that S is the range of a projection on L∞ (G) which commutes with translation if and only if S is the range of a projection on L∞ (G) which commutes with convolution. Our first purpose in this paper is to generalize Bekka’s results for a certain class of left Banach G-module. This result is used to show that G is amenable if and only if whenever X is a left Banach G-module and S is a weak*-closed right invariant subspace of X* which is complemented in X*, then S is the range of a projection on X* which commutes with convolution. Finally, we explore the link between the projections properties and amenability of group algebras.