On intersections of ranges of projections of norm one in Banach spaces

Abstract

In this short note we are interested in studying Banach spaces in which the range of a projection of norm one whose kernel is of finite dimension is the intersection of ranges of finitely many projections of norm one whose kernels are of dimension one. We show that for a certain class of Banach spaces X X , the natural duality between X X and X ∗ ∗ X^{\ast \ast } can be exploited when the range of the projection is of finite codimension. We show that if X ∗ X^\ast is isometric to L 1 ( μ ) L^1(\mu ) , then any central subspace of finite codimension is an intersection of central subspaces of codimension one. These results extend a recent result of Bandyopadhyay and Dutta which was proved for ranges of projections of norm one with finite dimensional kernel in continuous function spaces and unifies some earlier work of Baronti and Papini.