An Application of Orthoisomorphisms to Non-Commutative $L^p$-Isometries

We prove that if there exists an into linear isometry between non-commutative //-spaces then there exists an into Jordan * -isomorphism between underlying von Neumann algebras, as an application of Araki-Bunce-Wright's theorem concerning the characterization of orthogonality preserving positive maps between preduals. Moreover, we determine the structure of a linear non-commutative LMsometry when it is surjective and *-preserving.