Good Projections of Spaces of Vector Measures onto Subspaces of Bochner Integrable Functions

Abstract

We show that the complementability of L ₁ (μ, X) in cabv (μ, X) implies the complementability of L ₁ (μ, K(Z, X)) in cabv (μ, K(Z, X)), provided the projection from cabv (μ, X) onto L ₁ (μ, X) is "good",Z ^∗ is separable and K (<i Z, X</i) = L (<i Z, X</i). The projection got is also "good", so that it allows to construct a projection from the space L (L ₁ (μ), K (Z, X)) onto the subspace R (L ₁ (μ), K (Z, X)) of all representable operators