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

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.