  We prove that, for an action $\alpha \colon G \curvearrowright X$ of a
countably infinite discrete amenable group on a compact metric space, the small
boundary property is implied by uniform property $\Gamma$ of the Cartan
subalgebra $(C(X) \subseteq C(X) \rtimes_\alpha G)$. The reverse implication
has been demonstrated by Kerr and Szab\'o for free actions, from which we
obtain the equivalence of the two conditions in the free case. We moreover show
that, if $\alpha$ is free and minimal, then almost finiteness of $\alpha$ is
implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq
C(X) \rtimes_\alpha G)$. The reverse implication is due to Kerr, resulting in
the equivalence of these two properties as well. As an application, we prove
that if $\alpha \colon G \curvearrowright X$ and $\beta \colon H
\curvearrowright Y$ are free actions and $\alpha$ has the small boundary
property, then $\alpha \times \beta \colon G \times H \curvearrowright X \times
Y$ has the small boundary property. An analogous permanence property is
obtained for almost finiteness in case $\alpha$ and $\beta$ are free minimal
actions.
