Star order and topologies on von Neumann algebras

TL;DR

This paper investigates a topology generated by the star order on von Neumann algebras, showing it is finer than the sigma-strong* topology and comparable to the norm topology only in finite-dimensional cases.

Contribution

It introduces and analyzes a new topology based on the star order, comparing it with existing topologies on von Neumann algebras.

Findings

The star order topology is finer than the sigma-strong* topology.

The star order topology is comparable to the norm topology only in finite-dimensional von Neumann algebras.

The topology's properties depend on the dimensionality of the algebra.

Abstract

The goal of the paper is to study a topology generated by the star order on von Neumann algebras. In particular, it is proved that the order topology under investigation is finer than $\sigma$-strong* topology. On the other hand, we show that it is comparable with the norm topology if and only if the von Neumann algebra is finite-dimensional.