Some properties for certain generalized tracial approximated ${\mathrm{C}}^{\ast }$-algebras

In this paper, we introduce a class of generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^}$-algebraswhichhavetracially$$\mathcal{Z}$$-absorbing (tracial nuclear dimension at most $n$, $\mathrm{S}\mathrm{P}$ property, $m$-almost divisible, weakly $(m, n)$-divisible)\mathrm{.}Then$A$hastracially$$\mathcal{Z}$$-absorbing (tracial nuclear dimension at most $n$, $\mathrm{S}\mathrm{P}$ property, weakly $m$-almost divisible, secondly weakly $(m,n)$-divisible) for any simple unital ${\mathrm{C}}^{\ast }$-algebra $A$ in the class of this generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras. As an application, Let $A$ be an infinite dimensional unital simple ${\rm C^}$-algebra,andlet$B$beacentrallylargesubalgebraof$A$\mathrm{.}If$B$ is tracially $\mathcal{Z}$-absorbing, then $A$ is tracially $\mathcal{Z}$-absorbing. This result was obtained by Archey, Buck and Phillips in \cite{AJN}.