Hermitian operators and isometries on symmetric operator spaces

Abstract

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $\mathcal{H}$ equipped with a semifinite faithful normal trace $\tau$. Let $E(\mathcal{M},\tau )$ be a symmetric operator space affiliated with $\mathcal{M}$ , whose norm is order continuous and is not proportional to the Hilbertian norm ${\parallel \cdot \parallel }_2$ on $L_2(\mathcal{M},\tau )$. We obtain a general description of all bounded hermitian operators on $E(\mathcal{M},\tau )$. This is the first time that the description of hermitian operators on a symmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.