On the small boundary property, Z-stability, and Bauer simplexes

Let X be a compact metrizable space, and let Δ be a closed set of Borel probability measures on X. We study the small boundary property of the pair (X, Δ). In particular, it is shown that (X, Δ) has the small boundary property if it has a restricted version of property Gamma. As an application, it is shown that, if A is the crossed product C*-algebra C(X) Zᵈ, where (X, Zᵈ) is a free minimal topological dynamical system, or if A is an AH algebra with diagonal maps, then, A is Z-stable if the set of extreme tracial states is compact, regardless of its dimension.