Hermitian operators and isometries on symmetric operator spaces

TL;DR

This paper characterizes all bounded Hermitian operators on symmetric operator spaces affiliated with certain von Neumann algebras, resolving a long-standing open problem about isometries in noncommutative functional analysis.

Contribution

It provides the first comprehensive description of Hermitian operators on asymmetric noncommutative spaces for general von Neumann algebras, including non-hyperfinite cases.

Findings

Complete description of bounded Hermitian operators on symmetric operator spaces.

Resolution of a long-standing open problem on isometries in noncommutative spaces.

Unification of previous results in noncommutative geometry and operator theory.

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 $H$ 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 $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},\tau)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},\tau)$. This is the first time that the description of hermitian operators on asymmetric 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.