some properties for asymptotically tracially approximation of C*-algebras

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).