Isometric Structure in Noncommutative Symmetric Spaces

TL;DR

This paper characterizes isometries in noncommutative symmetric spaces, resolving longstanding open questions about their structure, and establishes conditions under which these spaces are uniquely determined by their isometric properties.

Contribution

It provides a complete description of isometries in noncommutative symmetric spaces, answering open problems from the 1980s and 2024, and characterizes when such spaces are uniquely determined by their symmetric structure.

Findings

Isometries are of elementary form in noncommutative symmetric spaces under certain conditions.

Noncommutative $L_p$-spaces have a unique symmetric structure up to isometries.

Spaces not isometric to $L_p$ over semifinite infinite traces are characterized.

Abstract

This is a systematic study of isometries between noncommutative symmetric spaces. Let $\mathcal{M}$ be a semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space $\mathcal{H}$ equipped with a semifinite faithful normal trace $\tau$. We show that for any noncommutative symmetric space corresponding to a symmetric function space $E(0,\infty)$ in the sense of Lindenstrauss--Tzafriri such that $\left\|\cdot\right\|_E\ne \lambda \left\|\cdot\right\|_{L_2}$, $\lambda\in \mathbb{R}_+$, any isometry on $E(\mathcal{M},\tau)$ is of elementary form. This answers a long-standing open question raised in the 1980s in the non-separable setting [Math. Z. 1989], while the case of separable symmetric function spaces was treated in [Huang \& Sukochev, JEMS, 2024]. As an application, we obtain a noncommutative Kalton--Randrianantoanina--Zaidenberg Theorem, providing a characterization of noncommutative $L_p$-spaces over finite von Neumann algebras and a necessary and sufficient condition for an operator on a noncommutative symmetric space to be an isometry. Having this at hand, we answer a question posed by Mityagin in 1970 [Uspehi Mat. Nauk] and its noncommutative counterpart by showing the any symmetric space $E(\mathcal{M},\tau)\ne L_p(\mathcal{M},\tau)$ over a noncommutative probability is not isometric to a symmetric space over a von Neumann algebra equipped with a semifinite infinite faithful normal trace. It is also shown that any noncommutative $L_p$-space, $1\le p<\infty$, affiliated with an atomless semifinite von Neumann algebra has a unique symmetric structure up to isometries. This contributes to the resolution of an isometric version of Pe\l czy\'nski's problem concerning the uniqueness of the symmetric structure in noncommutative symmetric spaces.