Perturbations of self-adjoint operators in semifinite von Neumann algebras: Kato-Rosenblum theorem

In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let M be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space H and let τ be a faithful normal semifinite tracial weight of M. Suppose that H and H1 are self-adjoint operators affiliated with M. We show that if H−H1 is in M∩L1(M,τ), then the norm absolutely continuous parts of H and H1 are unitarily equivalent. This implies that the real part of a non-normal hyponormal operator in M is not a perturbation by M∩L1(M,τ) of a diagonal operator.