Tracial approximation in simple C*-algebras

TL;DR

This paper explores the concept of tracial approximation in unital simple C*-algebras, establishing equivalences between classes with finite nuclear dimension and Z-stability, advancing understanding of their structural properties.

Contribution

It demonstrates that unital simple separable C*-algebras asymptotically tracially in finite nuclear dimension are equivalent to those asymptotically tracially in simple nuclear Z-stable algebras.

Findings

Equivalence of tracial approximation in finite nuclear dimension and Z-stability classes.

Characterization of unital simple C*-algebras via tracial approximation.

Enhanced understanding of structural properties of simple C*-algebras.

Abstract

We revisit the notion of tracial approximation for unital simple C*-algebras. We show that a unital simple separable C*-algebra A is asymptotically tracially in the class of C*-algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of simple nuclear Z-stable C*-algebras.