On the small boundary property, $\mathcal Z$-absorption, and Bauer simplexes
George A. Elliott, Zhuang Niu
TL;DR
This work extends the small boundary property (SBP) to the pair $(X, \Delta)$, where $X$ is compact and metrizable and $\Delta$ is a closed set of Borel probability measures. It characterizes SBP via a trace-based $2$-norm criterion and a real rank zero condition on the sequence algebra, and introduces a strengthened uniform Gamma-type notion via approximate divisibility. The authors prove that approximate divisibility implies SBP, and in Bauer simplex scenarios they show SBP holds for crossed products $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ and for AH algebras with diagonal maps, yielding $\mathcal{Z}$-stability without resorting to W*-bundle machinery. Consequently, these results establish $\mathcal{Z}$-absorption for a broad class of dynamical and diagonal AH algebras under compact trace boundaries, expanding the toolkit for verifying $\mathcal{Z}$-stability in noncommutative topology.
Abstract
Let $X$ be a compact metrizable space, and let $Δ$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X, Δ)$. In particular, it is shown that $(X, Δ)$ 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 $\mathrm{C}(X)\rtimes\mathbb Z^d$, where $(X, \mathbb Z^d)$ is a free minimal topological dynamical system, or if $A$ is an AH algebra with diagonal maps, then, $A$ is $\mathcal Z$-stable if the set of extreme tracial states is compact, regardless of its dimension.