Preduals for spaces of operators involving Hilbert spaces and trace-class operators

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(S1), the algebra of operators on the space of trace-class operators, correspond to isometric preduals of S1 itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of S1 making its multiplication separately weak⁎ continuous.