Logarithmic submajorisation and order-preserving linear isometries

Let $\mathcal{E}$ and $\mathcal{F}$ be symmetrically $\mathrm{\Delta }$-normed (in particular, quasi-normed) operator spaces affiliated with semifinite von Neumann algebras ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$, respectively. We establish a noncommutative version of Abramovich’s theorem \cite{A1983}, which provides the general form of normal order-preserving linear operators $T:\mathcal{E}\mathcal{F}$ having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem \cite{Huijsmans_W}. By establishing the disjointness preserving property for an order-preserving isometry $T:\mathcal{E}\mathcal{F}$, we obtain the existence of a Jordan $\ast$-monomorphism from ${\mathcal{M}}_1$ into ${\mathcal{M}}_2$ and the general form of this isometry, which extends and complements a number of existing results. In particular, we fully resolve the case when $\mathcal{F}$ is the predual of ${\mathcal{M}}_2$ and other untreated cases in [Sukochev-Veksler, IEOT, 2018].