A limit in probability in a W-algebra is unique

Let be the space of self-adjoint Segal measurable operators affiliated to a W^*-algebra (for x=∫λe_x(dλ), xe_x((−∞, −n)(n, ∞)) is a finite projection in for n large enough). The limit in probability is unique in . x_n → x in probability e_nProj;e_n→1 strongly and (x_n − x) e_n → 0. The following proposition proves to be important in the investigation of . If 1 − ƒ is a finite projection in and 1 − e_n → 0 strongly, e_nProj, then strongly.