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

Abstract

We prove that if there exists an into linear isometry between non-commutative $L^p$-spaces then there exists an into Jordan $\ast$ -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 $L^p$-isometry when it is surjective and $\ast$-preserving.