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

Let X be a compact metrizable space, and let be a closed set of Borel probability measures on X. We study the small boundary property of the pair (X, ). In particular, it is shown that (X, ) 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.