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 $A=\mathrm{C}(X)⋊\Gamma$, 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 $(X,\Gamma )$.

This is a further revision of version 2. Property (C) of the Cartan pair of C*-algebras is equivalent to the small boundary property of the dynamical system. But it is still unknown if it holds for Jiang-Su-absorbing C*-algebras