Logarithmic submajorisation and order-preserving linear isometries

Abstract

Let E and F be noncommutative operator spaces affiliated with semifinite von Neumann algebras M1 and M2, respectively. We establish a noncommutative version of Abramovich's theorem [1], which provides the general form of normal order-preserving linear operators T:E⟶intoF having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem [44]. By establishing the disjointness preserving property for an order-preserving isometry T:E→F from a noncommutative symmetrically Δ-normed (in particular, quasi-normed) space into another, we obtain the existence of a Jordan ⁎-monomorphism from M1 into M2 and the general form of this isometry, which extends and complements a number of existing results such as [12, Theorem 1], [67, Corollary 1], [72, Theorem 2] and [15, Theorem 3.1]. In particular, we fully resolve the case when F is the predual of M2 and other untreated cases in [78].