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

Abstract

We introduce Property (C) for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, which is a version of the relative comparison property using almost normalizers. In the case that $D = \mathrm{C}(X)$ and $A = \mathrm{C}(X) \rtimes\Gamma$, where $(X, \Gamma)$ is a free and minimal dynamical system with uniform Rokhlin property, it turns out that this Property (C) is equivalent to the small boundary property of $(X, \Gamma)$.