Strict comparison and Z-absorption of nuclear C*-algebras

TL;DR

This paper establishes the equivalence of Z-absorption, strict comparison, and property (SI) for certain nuclear C*-algebras, and shows that those with tracial rank zero are approximately divisible and Z-absorbing.

Contribution

It proves the equivalence of key properties in nuclear C*-algebras with finitely many extremal traces and demonstrates that tracial rank zero implies Z-absorption.

Findings

Z-absorption, strict comparison, and property (SI) are equivalent for specified C*-algebras.

Unital separable simple nuclear C*-algebras with tracial rank zero are approximately divisible.

Such algebras are Z-absorbing.

Abstract

For any unital separable simple infinite-dimensional nuclear C*-algebra with finitely many extremal traces, we prove that Z-absorption, strict comparison, and property (SI) are equivalent. We also show that any unital separable simple nuclear C*-algebra with tracial rank zero is approximately divisible, and hence is Z-absorbing.