Almost elementary groupoid models for $C^*$-algebras

Xin Ma, Jianchao Wu
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Abstract

The notion of almost elementariness for a locally compact Hausdorff \'{e}tale groupoid $\mathcal{G}$ with a compact unit space was introduced by the authors as a sufficient condition ensuring the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ is (tracially) $\mathcal{Z}$-stable and thus classifiable under additional natural assumption. In this paper, we explore the converse direction and show that many groupoids in the literature serving as models for classifiable $C^*$-algebras are almost elementary. In particular, for a large class $\mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with $\operatorname{Ell}(A)\in \mathcal{C}$, we show that $A$ is classifiable if and only if $A$ possesses a minimal, effective, amenable, second countable, almost elementary groupoid model, which leads to a groupoid-theoretic characterization of classifiability of $C^*$-algebras with certain Elliott invariants. Moreover, we build a connection between almost elementariness and pure infiniteness for groupoids and study obstructions to obtaining a transformation groupoid model for the Jiang-Su algebra $\mathcal{Z}$.
C^*$ 算法的几乎基本群模型
作者提出了一个概念,即对于具有紧凑单位空间的局部紧凑 Hausdorff \'{e}talegroupoid $\mathcal{G}$ 来说,几乎元素性是一个充分条件,可以确保还原的基元 $C^*$-algebra$C^*_r(\mathcal{G})$ 是(tracially)$\mathcal{Z}$ 稳定的,从而在额外的自然假设下是可分类的。在本文中,我们探索了对话方向,并证明了文献中许多作为可分类 $C^*$ 算法模型的基元几乎都是基本的。特别是,对于埃利奥特不变式的大类 $\mathcal{C}$ 和在 \mathcal{C}$ 中具有$\operatorname{Ell}(A)\的$C^*$-代数 $A$,我们证明,如果且只有当 $A$ 具有一个最小值时,$A$ 才是可分类的、我们证明,只有当且仅当 $A$ 拥有一个最小的、有效的、友好的、第二可数的、几乎是元素的类群模型时,$A$ 才是可分类的。此外,我们还在几乎元元性和纯无限性之间建立了联系,并研究了江苏代数 $\mathcal{Z}$ 获得变换群元模型的障碍。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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