Preduals and complementation of spaces of bounded linear operators

Abstract

For Banach spaces X and Y, we establish a natural bijection between preduals of Y and preduals of L(X,Y) that respect the right L(X)-module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement Y in its bidual and L(X)-linear projections that complement L(X,Y) in its bidual. It follows that Y is complemented in its bidual if and only if L(X,Y) is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.