Classically normal pure states

TL;DR

This paper characterizes when separably represented von Neumann algebras have classically normal, singular pure states, linking their existence to the presence of certain types of factors and the continuum hypothesis.

Contribution

It provides a characterization of classically normal, singular pure states in separably represented von Neumann algebras under the continuum hypothesis.

Findings

Existence of classically normal, singular pure states is equivalent to the presence of a specific central projection.

Such states exist iff the algebra contains a factor of type I_, II, or III.

The result depends on the continuum hypothesis.

Abstract

A pure state f of a von Neumann algebra M is called classically normal if f is normal on any von Neumann subalgebra of M on which f is multiplicative. Assuming the continuum hypothesis, a separably represented von Neumann algebra M has classically normal, singular pure states iff there is a central projection p in M such that Mp is a factor of type I_\infty, II, or III.