Positive Linear Isometries in Symmetric Operator Spaces

Let (M, tau) and (N, nu) be semifinite von Neumann algebras equipped with faithful normal semifinite traces and let E(M, tau) and F(N, nu) be symmetric operator spaces associated with these algebras. We provide a sufficient condition on the norm of the space F(N, nu) guaranteeing that every positive linear isometry T : E(M, tau) ->(into) F(N, nu) is "disjointness preserving" in the sense that T(x)T(y) = 0 provided that xy = 0, 0 <= x, y is an element of E(M, tau). This fact, in turn, allows us to describe the general form of such isometries.