On tracial $\mathcal Z$-stability of simple non-unital C*-algebras

TL;DR

This paper extends the concept of tracial al al-stability to non-unital C*-algebras and proves its equivalence to al al-stability in separable simple nuclear cases.

Contribution

It generalizes the notion of tracial al al-stability to non-unital algebras and establishes its equivalence with al al-stability for a broad class of C*-algebras.

Findings

Tracial al al-stability is equivalent to al al-stability in separable simple nuclear C*-algebras.

The notion is extended beyond unital C*-algebras.

The equivalence simplifies classification of these algebras.

Abstract

We investigate the notion of tracial $\mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $\mathcal Z$-stability in the class of separable simple nuclear C*-algebras.