Generalized Wave Operators in von Neumann Algebras

TL;DR

This paper introduces new types of generalized wave operators within von Neumann algebras and investigates their relationships with existing wave operators, expanding the theoretical framework of operator theory.

Contribution

It defines generalized weak, weak abelian, and stationary wave operators in von Neumann algebras and explores their interrelations with classical wave operators.

Findings

Defined generalized wave operators in von Neumann algebras.

Established relations among various generalized wave operators.

Extended the theoretical understanding of wave operators in operator 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 $\tau$ 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 $W_{\pm}.$