Comparison properties of asymptotically tracially approximation C*-algebras

We show that the following properties of the C*-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class of asymptotically tracially in $\mathcal{P}$: $(1)$ $β$-comparison (in the sense of Kirchberg and Rørdam), $(2)$ $n$-comparison (in the sense of Winter).

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