Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability

Inspired by Kerr’s work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^{\ast }$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^{\ast }$-algebra $(C(X)\subseteq C(X)⋊G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.