On the small boundary property and $\mathcal Z$-absorption, II

Abstract: Consider a minimal and free topological dynamical system $(X, \mathbb Z^d)$. It is shown that zero mean dimension of $(X, \mathbb Z^d)$ is characterized by $\mathcal Z$-absorption of the crossed product C*-algebra $A=\mathrm{C}(X) \rtimes \mathbb Z^d$, where $\mathcal Z$ is the Jiang-Su algebra. In fact, among other conditions, the following are shown to be equivalent: (1) $(X, \mathbb Z^d)$ has the small boundary property. (2) $A \cong A \otimes \mathcal Z$. (3) $A$ has uniform property $\Gamma$. (4) $l^\infty(A)/J_{2, \omega, \mathrm{T}(A)}$ has real rank zero. The same statement also holds for unital simple AH algebras with diagonal maps.