Non unital generalized tracially approximated C*-algebras

Abstract

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