Maximally unitarily mixed states on a C*-algebra

TL;DR

This paper studies the structure of maximally mixed states in C*-algebras, revealing differences from von Neumann algebras and providing descriptions of their closures and special cases.

Contribution

It extends previous work by characterizing the weak* closure of maximally mixed states in C*-algebras and explores specific cases like those with the Dixmier property.

Findings

The set of maximally mixed states may not be weak* closed in C*-algebras.

The weak* closure can be described using tracial states and simple traceless quotients.

For certain C*-algebras, the set of maximally mixed states is explicitly characterized.

Abstract

We investigate the set of maximally mixed states of a C*-algebra, extending previous work by Alberti on von Neumann algebras. We show that, unlike for von Neumann algebras, the set of maximally mixed states of a C*-algebra may fail to be weak* closed. We obtain, however, a concrete description of the weak* closure of this set, in terms of tracial states and states which factor through simple traceless quotients. For C*-algebras with the Dixmier property or with Hausdorff primitive spectrum we are able to advance our investigations further. In the latter case we obtain a concrete description of the set of maximally mixed states in terms of traces and extensions of the states of a closed two-sided ideal. We pose several questions.