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.