Generalized Wave Operators in von Neumann Algebras

Abstract

Let $\mathcal{M}\subseteq\mathcal{B}\left( \mathcal{H}\right) $ be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weight $τ$ where $\mathcal{B}\left( \mathcal{H}\right) $ is the set of all bounded linear operators on Hilbert space $\mathcal{H}.$ The main purpose of this article is to introduce generalized weak wave operators $\widetilde{W}_{\pm}$, generalized weak abelian wave operators $\widetilde{\mathfrak{U}}_{\pm}$ and generalized stationary wave operators $\mathcal{U}_{\pm}$ in $\mathcal{M}$ and then to explore the relation among $\widetilde{W}_{\pm},$ $\widetilde{\mathfrak{U}% }_{\pm}$, $\mathcal{U}_{\pm}$ and generalized wave operators $W_{\pm}.$