A stationary approach for the Kato–Rosenblum theorem in von Neumann algebras

Abstract

Let \({\mathcal {M}}\) be a countable decomposable properly infinite semifinite von Neumann algebra acting on a Hilbert space \({\mathcal {H}}.\) An analogue of the Kato–Rosenblum theorem in \({\mathcal {M}}\) has been proved in Li et al. (J Funct Anal 275(2):259–287, 2018) by showing the existence of generalized wave operators. It is well known that there are two typical approaches to show the existence of wave operators in the scattering theory. One is called time-dependent approach and another is called stationary approach. The main purpose of this article is to exhibit the stationary scattering theory in \({\mathcal {M}}\) and then to obtain the Kato–Rosenblum theorem in \({\mathcal {M}}\) by the stationary approach instead of a time-dependent approach in Li et al. (2018).