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

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 $\mathcal{Z}$.