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

Abstract

In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $\tau$ be a faithful normal semifinite tracial weight of $\mathcal M$. Suppose that $H$ and $H_1$ are self-adjoint operators affiliated with $\mathcal{M}$. We show that if $H-H_1$ is in $\mathcal{M}\cap L^{1}\left(\mathcal{M},\tau\right)$, then the ${norm}$ absolutely continuous parts of $H$ and $H_1$ are unitarily equivalent. This implies that the real part of a non-normal hyponormal operator in $\mathcal M$ is not a perturbation by $\mathcal{M}\cap L^{1}\left(\mathcal{M},\tau\right)$ of a diagonal operator. Meanwhile, for $n\ge 2$ and $1\leq p<n$, by modifying Voiculescu's invariant we give examples of commuting $n$-tuples of self-adjoint operators in $\mathcal{M}$ that are not arbitrarily small perturbations of commuting diagonal operators modulo $\mathcal{M}\cap L^{p}\left(\mathcal{M},\tau\right)$.