Tracial approximation and ${\cal Z}$-stability

TL;DR

This paper establishes a characterization of ${ m Z}$-stability for certain simple C*-algebras based on strict comparison and stable rank one, extending the results to non-unital cases.

Contribution

It provides a new if-and-only-if criterion for ${ m Z}$-stability in simple C*-algebras with specific tracial boundary conditions, including non-unital cases.

Findings

${ m Z}$-stability characterized by strict comparison and stable rank one.

Extension of results to non-unital C*-algebras.

Applicable to a broad class of simple amenable C*-algebras.

Abstract

Let $A$ be a unital separable non-elementary amenable simple stably finite C*-algebra such that its tracial state space has a $\sigma$-compact countable-dimensional extremal boundary. We show that $A$ is ${\cal Z}$-stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).