Isometric Structure in Noncommutative Symmetric Spaces

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 $τ$. 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 $$\left|\cdot\right|<em>E\ne λ\left|\cdot\right|</em>{L_2}$$, $λ\in \mathbb{R}_+$, any isometry on $E(\mathcal{M},τ)$ 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 $E(\mathcal{M},τ)\ne L_p(\mathcal{M},τ)$ 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, $1\le p&lt;\infty$, 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 Pełczyński's problem concerning the uniqueness of the symmetric structure in noncommutative symmetric spaces.