Pure C*-algebras
Abstract
We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and with very mild comparison is pure, and, as a result, has strict comparison. Furthermore, sufficiently non-commutative C*-algebras of stable rank one and with weak comparison are likewise pure. We also show that adequately non-elementary C*-algebras with finite nuclear dimension are pure, which leads to the verification of the non-simple Toms-Winter conjecture for a large class of C*-algebras.
Keywords
Cite
@article{arxiv.2406.11052,
title = {Pure C*-algebras},
author = {Ramon Antoine and Francesc Perera and Hannes Thiel and Eduard Vilalta},
journal= {arXiv preprint arXiv:2406.11052},
year = {2024}
}
Comments
Improvements throughout; added applications to C*-simple groups and Villadsen algebras; the paper has now been submitted