Positive Linear Isometries in Symmetric Operator Spaces

Abstract

Let $(\mathcal {M},\tau )$ and $(\mathcal {N},\nu )$ be semifinite von Neumann algebras equipped with faithful normal semifinite traces and let $E(\mathcal {M},\tau )$ and $F(\mathcal {N},\nu )$ be symmetric operator spaces associated with these algebras. We provide a sufficient condition on the norm of the space $F(\mathcal {N},\nu )$ guaranteeing that every positive linear isometry $T:E(\mathcal {M},\tau ){\mathop {\longrightarrow }\limits ^{into}} F(\mathcal {N},\nu )$ is “disjointness preserving” in the sense that $T(x)T(y)=0$ provided that $xy=0$, $0\le x,y\in E(\mathcal {M},\tau )$. This fact, in turn, allows us to describe the general form of such isometries.