This is an announcement for the paper “Preduals and complementation of spaces of bounded linear operators” by Eusebio Gardellahttps://arxiv.org/find/math/1/au:+Gardella_E/0/1/0/all/0/1, Hannes Thielhttps://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1.

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 projections that complement $L(X,Y)$ in its bidual as a right $L(X)$-module. It follows that $Y$ is complemented in its bidual if and only if $L(X,Y)$ is complemented in its bidual (either as a module or as a Banach space).

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.05326