Preduals and complementation of spaces of bounded linear operators

TL;DR

This paper explores the structure of preduals and complemented subspaces of spaces of bounded linear operators, establishing new bijections and removing previous restrictions, with implications for dual Banach algebras.

Contribution

It introduces natural bijections between preduals of Y and L(X,Y), and between complemented projections, removing the approximation property condition.

Findings

Unique predual making L(X) a dual Banach algebra when X is reflexive

Bijection between projections complementing Y and those complementing L(X,Y)

Y is complemented in its bidual iff L(X,Y) is complemented in its bidual

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.