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

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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. Meanwhile, for n≥ 2 and 1≤ p<n, by modifying Voiculescu's invariant we give examples of commuting n-tuples of self-adjoint operators in M that are not arbitrarily small perturbations of commuting diagonal operators modulo M∩ Lᵖ(M,τ).