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

## TL;DR

This paper explores the structure of preduals of operator spaces involving Hilbert spaces and trace-class operators, establishing correspondences and uniqueness results for their isometric preduals.

## Contribution

It introduces a natural correspondence between isometric preduals of operator spaces and their underlying spaces, extending the understanding of preduals in the context of Hilbert and trace-class operators.

## Key findings

- Isometric preduals of $\\mathcal{L}(H,Y)$ correspond to those of $Y$.
- Preduals of $\\mathcal{L}(\mathcal{S}_1)$ relate to preduals of $\mathcal{S}_1$.
- The compact operators are the unique predual of $\mathcal{S}_1$ with separately weak* continuous multiplication.

## 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.