Uniform property and the small boundary property

We prove that, for a free action G X of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property of the Cartan subalgebra (C (X) C (X) _ G). The reverse implication has been demonstrated by Kerr and Szab\'o, from which we obtain that these two conditions are equivalent. We moreover show that, if is also minimal, then almost finiteness of is implied by tracial Z-stability of the subalgebra (C (X) C (X) _ 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 G X and H Y are free actions and has the small boundary property, then the product G H X Y has the small boundary property. An analogous permanence property is obtained for almost finiteness in the case of free minimal actions.