Non unital generalized tracially approximated C*-algebras

Let $\mathrm{\Omega }$ be a class of ${\mathrm{C}}^{\ast }$-algebras. In this paper, we study a class of not necessarily unital generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras, and the class of simple ${\mathrm{C}}^{\ast }$-algebras which can be generally tracially approximated by ${\mathrm{C}}^{\ast }$-algebras in $\mathrm{\Omega }$, denoted by $\mathrm{g}\mathrm{T}\mathrm{A}\mathrm{\Omega }$. Let $\mathrm{\Omega }$ be a class of unital ${\mathrm{C}}^{\ast }$-algebras and let $A$ be a simple unital ${\mathrm{C}}^{\ast }$-algebra. Then $A\in \mathrm{g}\mathrm{T}\mathrm{A}\mathrm{\Omega }$, if, and only if, $A\in \mathrm{W}\mathrm{T}\mathrm{A}\mathrm{\Omega }$ (where $\mathrm{T}\mathrm{A}\mathrm{\Omega }$ is the class of weakly tracially approximable unital ${\mathrm{C}}^{\ast }$-algebras introduced by Elliott, Fan, and Fang).Consider the class of ${\mathrm{C}}^{\ast }$-algebras which are tracially $\mathcal{Z}$-absorbing (or are of tracial nuclear dimension at most $n$, or are $m$-almost divisible, or have the property $\mathrm{S}\mathrm{P}$). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, is weakly ($n,m$)-almost divisible, has the property $\mathrm{S}\mathrm{P}$) for any simple ${\mathrm{C}}^{\ast }$-algebra $A$ in the corresponding class of generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras.