some properties for asymptotically tracially approximation of C*-algebras

TL;DR

This paper investigates how certain properties of unital C*-algebras, such as real rank zero and specific comparison radii, are preserved within classes of asymptotically tracially approximated algebras, with implications for their tracial ranks.

Contribution

It demonstrates that properties like real rank zero and the radius of comparison are inherited by simple unital C*-algebras in the class TΩ, expanding understanding of their structural characteristics.

Findings

Real rank zero is inherited in TΩ.

C*-algebras with radius of comparison n are inherited in TΩ.

Algebras with certain tracial ranks are preserved in TΩ.

Abstract

Let $\Omega$ be a class of unital $\rm C^{*}$-algebras. The class of ${\rm C^*}$-algebras which are asymptotical tracially in $\Omega$, denoted by ${\rm AT}\Omega$. In this paper, we will show that the following class of ${\rm C^*}$-algebras in the class $\Omega$ are inherited by simple unital ${\rm C^*}$-algebras in the class $\rm T\Omega$ $(1)$ the class of real rank zero ${\rm C^*}$-algebras, $(2)$ the class of ${\rm C^*}$-algebras with the radius of comparison $n$, and $(3)$ the class of $\rm AT\Omega$. As an application, let $\Omega$ be a class of unital ${\rm C^*}$-algebras which have generalized tracial rank at most one (or has tracial topological rank zero, or has tracial topological rank one). Let $A$ be a unital separable simple ${\rm C^*}$-algebra such that $A\in \rm AT\Omega$, then $A$ has generalized tracial rank at most one (or has tracial topological rank zero, or has tracial topological rank one).