Comparison of topologies on *-algebras of locally measurable operators

TL;DR

This paper investigates the conditions under which the locally measure topology on the algebra of locally measurable operators coincides with the order topology, focusing on properties of the underlying von Neumann algebra.

Contribution

It characterizes when the locally measure topology matches the order topology on self-adjoint operators, specifically for σ-finite and finite von Neumann algebras.

Findings

The topology $t(\mathcal{M})$ coincides with the $(o)$-topology if and only if $\mathcal{M}$ is $\sigma$-finite and finite.

Relationships between $t(\mathcal{M})$ and topologies from faithful normal semifinite traces are established.

Conditions for topological equivalences are precisely characterized for different classes of von Neumann algebras.

Abstract

We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if and only if the algebra $\mathcal{M}$ is $\sigma$-finite and a finite algebra. We study relationships between the topology $t(\mathcal{M})$ and various topologies generated by faithful normal semifinite traces on $\mathcal{M}$.