The order topology for a von Neumann algebra

TL;DR

This paper investigates the order topology and its sequential variant on key posets within a von Neumann algebra, analyzing their properties and relationships to standard topologies and the algebraic structure.

Contribution

It introduces and studies the order topology on self-adjoint parts and projections of von Neumann algebras, comparing it with classical topologies and exploring its algebraic implications.

Findings

Order topology on $M_{sa}$ and $P(M)$ is characterized and compared to standard topologies.

Sequential order topology differs from the order topology in certain cases.

Properties of the order topology reflect the structure of the von Neumann algebra.

Abstract

The order topology $\tau_o(P)$ (resp. the sequential order topology $\tau_{os}(P)$) on a poset $P$ is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra $M$ we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M_{sa}^1$, and the projection lattice $P(M)$. We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on $M$, and relate the properties of the order topology to the underlying operator-algebraic structure of $M$.