some properties for asymptotically tracially approximation of C*-algebras

Abstract

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

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