Comparison properties of asymptotically tracially approximation C*-algebras

TL;DR

This paper investigates how certain comparison properties, specifically $eta$-comparison and $n$-comparison, are inherited by simple unital C*-algebras that are asymptotically tracially approximated by a class $f C*-algebras.

Contribution

It demonstrates that properties like $eta$-comparison and $n$-comparison are preserved in simple unital C*-algebras within the asymptotic tracial approximation framework.

Findings

Inheritance of $eta$-comparison in asymptotically tracially approximated algebras.

Inheritance of $n$-comparison in asymptotically tracially approximated algebras.

Extension of comparison properties to broader classes of C*-algebras.

Abstract

We show that the following properties of the C*-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class of asymptotically tracially in $\mathcal{P}$: $(1)$ $\beta$-comparison (in the sense of Kirchberg and R{\o}rdam), $(2)$ $n$-comparison (in the sense of Winter).