Topologies on Central Extensions of Von Neumann Algebras

TL;DR

This paper studies the topology on the central extension of a von Neumann algebra, establishing conditions under which it coincides with other known topologies, depending on the algebra's type and summands.

Contribution

It introduces a new topology on the central extension of von Neumann algebras and characterizes when it matches convergence in measure and the order topology.

Findings

Topology $t_c(M)$ coincides with local measure convergence iff $M$ has no type II summands.

On self-adjoint elements, $t_c(M)$ matches the order topology iff $M$ is a $\sigma$-finite type I$_{fin}$ algebra.

Provides a detailed analysis of topologies on central extensions of von Neumann algebras.

Abstract

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ We introduce the topology $t_c(M)$ on $E(M)$ generated by a center-valued norm and prove that it coincides with the topology of convergence locally in measure on $E(M)$ if and only if $M$ does not have direct summands of type II. We also show that $t_c(M)$ restricted on the set $E(M)_h$ of self-adjoint elements of $E(M)$ coincides with the order topology on $E(M)_h$ if and only if $M$ is a $\sigma$-finite type I$_{fin}$ von Neumann algebra.