LAUSR

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