This is an announcement for the paper “Preduals for spaces of operators involving Hilbert spaces and trace-class operators” by Hannes Thielhttps://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1.
Abstract: Continuing the study of preduals of spaces $L(H, Y)$ of bounded, linear maps, we consider the situation that $H$ is a Hilbert space. We establish a natural correspondence between isometric preduals of $L(H, Y)$ and isometric preduals of $Y$. The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements $L(H, Y)$ in its bidual is automatically a right $L(H)$-module map. As an application, we show that isometric preduals of $L(S_1)$, the algebra of operators on the space of trace-class operators, correspond to isometric preduals of $S_1$ itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of $S_1$ making its multiplication separately weak* continuous.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.01169