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

Abstract

We introduce the Property (C) for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, a version of the relative comparison property using almost normalizers. Under the assumption of this property, the $\mathcal Z$-absorption of $A$ is shown to imply the small boundary property of $(D, \mathrm{T}(A)|_D)$, where $A =\mathrm{C}(X) \rtimes \mathbb Z^d $ and $D = \mathrm{C}(X)$.