In this paper, we introduce a class of non-unital tracial approximation ${ \rm C^*}$-algebras. Consider the class of ${ \rm C^*}$-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 ${ \rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${ \rm C^*}$-algebras.
Categories: math.OA