Non-unital tracially approximated ${\mathrm{C}}^{\ast }$-algebras

In this paper, we introduce a class of non-unital tracial approximation ${\mathrm{C}}^{\ast }$-algebras. Consider the class of ${\mathrm{C}}^{\ast }$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips’s simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo’s tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\mathrm{C}}^{\ast }$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\mathrm{C}}^{\ast }$-algebras.