Pure C*-algebras

TL;DR

This paper establishes broad conditions under which C*-algebras are pure, connecting properties like comparison, divisibility, and nuclear dimension, and confirming the Toms-Winter conjecture for many cases.

Contribution

It proves that purity follows from weak comparison and divisibility, and verifies the Toms-Winter conjecture for a large class of C*-algebras.

Findings

Pure C*-algebras form a robust class under weak comparison.

Simplicity and mild comparison imply purity and strict comparison.

Finite nuclear dimension implies purity, supporting the Toms-Winter conjecture.

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.