Disjointness-preserving mappings on Calkin operator spaces and positive isometries

Abstract

Let $E(\mathcal{M},τ)$ and $F(\mathcal{M},τ)$ be two Calkin operator spaces affiliated with a semifinite von Neumann algebra $\mathcal{M}$ equipped with a semifinite faithful normal trace $τ$. We show that if $\mathcal{M}$ is atomless, $τ$ is finite, and $E(v,τ)\not\subseteq F(\mathcal{M},τ)$, then every order-measure continuous and disjointness-preserving mapping $T:E(\mathcal{M},τ)\xrightarrow{\rm into} F(\mathcal{M},τ)$ is identical to the zero mapping, which establishes a noncommutative version of Abramovich's theorem. We also show that every positive isometry $T$ from a normed $\mathcal{M}$-bimodule $E(\mathcal{M},τ)$ of $τ$-measurable operators into another $F(\mathcal{M},τ)$ preserves disjointness provided that the norm of $F(\mathcal{M},τ)$ is strictly monotone. As an application, we obtain the general form of $T$, which extends and unifies several results due to Abramovich, de Jager, Conradie, Veksler and Sukochev et al. \cite{SV,HSZ20,Abra1991,vek,dC20}.