Disjointness-preserving mappings on Calkin operator spaces and positive isometries

Abstract

Let $E(\mathcal{M},\tau )$ and $F(\mathcal{M},\tau )$ be two Calkin operator spaces affiliated with a semifinite von Neumann algebra $\mathcal{M}$ equipped with a semifinite faithful normal trace $\tau$. We show that if $\mathcal{M}$ is atomless, $\tau$ is finite, and $E(v,\tau )⊈F(\mathcal{M},\tau )$, then every order-measure continuous and disjointness-preserving mapping $T:E(\mathcal{M},\tau )\stackrel{\mathrm{into}}{\to }F(\mathcal{M},\tau )$ 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 $E(\mathcal{M},\tau )$ of $\tau$-measurable operators into another $F(\mathcal{M},\tau )$ preserves disjointness provided that the norm of $F(\mathcal{M},\tau )$ 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}.