几乎基本平展群胚
Almost elementary étale groupoids
AI总结：
本研究引入几乎基本性作为平展群胚的新有限逼近性质，证明极小几乎基本群胚的约化C*-代数为$\boldsymbol{\text{Z}}$-稳定，还探讨了其与群胚相关性质的联系。
摘要：
受Matui与Kerr关于几乎有限性研究的启发，我们为（可能非宽的）平展群胚引入了一种新的有限逼近性质，称为几乎基本性，该性质统一并推广了几乎有限性与纯无限性。此性质可作为C*-代数正则性质的动力学类比。为支撑这一观点，我们证明极小几乎基本群胚可产生迹类$\boldsymbol{\text{Z}}$-稳定约化群胚C*-代数。由此作为推论，我们得到Matui意义下所有极小可和第二可数几乎有限群胚的约化C*-代数均为$\boldsymbol{\text{Z}}$-稳定，因而可由Elliott不变量分类。几乎基本性定义的两个基础要素是群胚中的城堡与群胚次等价，二者在本研究中得到了广泛发展。值得注意的是，基于我们灵活的城堡框架，我们引入了城堡嵌套技术，该技术在主定理的证明中发挥关键作用。除了关于迹类$\boldsymbol{\text{Z}}$-稳定性的主定理外，我们还深入探讨了几乎基本性与群胚其他性质的关系，如有效性、群胚小边界性质、群胚严格比较以及Matui和Kerr的几乎有限性概念。
英文摘要：
Motivated by Matui and Kerr's work on almost finiteness, we introduce a new finite approximation property for (possibly non-ample) étale groupoids called {almost elementariness}, which unifies and generalizes both almost finiteness and pure infiniteness. This property serves as a dynamical analogue of regularity properties of $C^*$-algebras. In support of this view, we prove that minimal almost elementary groupoids yield tracially $\mathcal{Z}$-stable reduced groupoid $C^*$-algebras. Consequently, we obtain as a corollary that the reduced $C^*$-algebras of all minimal amenable second countable almost finite groupoids in Matui's sense are $\mathcal{Z}$-stable and thus classifiable by the Elliott invariants. Two basic ingredients underlying the definition of almost elementariness are castles in groupoids and groupoid subequivalence, both of which are developed extensively in this work. Notably, building on our flexible framework of castles, we introduce the technique of nesting of castles, which play a key role in the proof of our main theorem. Besides our main theorem on tracial $\mathcal{Z}$-stability, we also discuss in depth the relations between almost elementariness and other properties for groupoids such as effectiveness, the groupoid small boundary property, groupoid strict comparison and Matui's and Kerr's notions of almost finiteness.
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