Title: Preduals for spaces of operators involving Hilbert spaces and trace-class operators,
Abstract: Continuing the study of preduals of spaces $\mathcal{L}(H,Y)$ of bounded,
linear maps, we consider the situation that $H$ is a Hilbert space. We
establish a natural correspondence between isometric preduals of
$\mathcal{L}(H,Y)$ and isometric preduals of $Y$.
The main ingredient is a Tomiyama-type result which shows that every
contractive projection that complements $\mathcal{L}(H,Y)$ in its bidual is
automatically a right $\mathcal{L}(H)$-module map.
As an application, we show that isometric preduals of
$\mathcal{L}(\mathcal{S}_1)$, the algebra of operators on the space of
trace-class operators, correspond to isometric preduals of $\mathcal{S}_1$
itself (and there is an abundance of them). On the other hand, the compact
operators are the unique predual of $\mathcal{S}_1$ making its multiplication
separately weak* continuous.