Uniform property $Γ$ and the small boundary property
Grigoris Kopsacheilis, Hung-Chang Liao, Aaron Tikuisis, Andrea Vaccaro
TL;DR
The paper establishes a precise equivalence for free, amenable-group actions: the small boundary property of the action $\alpha$ is equivalent to the Cartan inclusion $\big(C(X) \subseteq C(X) \rtimes_\alpha G\big)$ having uniform property $Γ$. This deepens the connection between dynamical regularity notions and $C^*$-algebraic regularity, linking SBP with central sequence structure via complemented partitions of unity (CPoU). Moreover, the authors show that, under minimality, almost finiteness corresponds to tracial $\mathcal{Z}$-stability, and that these correspondences extend to permanence results for products of actions. The work also clarifies relationships among dynamical properties (e.g., dynamical comparison) and $C^*$-algebraic regularity, providing tools for analyzing crossed products in terms of Cartan subalgebras and central sequences. Collectively, the results yield a robust framework to transfer regularity phenomena between topological dynamics and $C^*$-algebra theory, with concrete implications for product actions and mean-dimension considerations.
Abstract
We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions.