2024/07/07 by Ma, Xin, Wu, Jianchao · 1 citation
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2407.05251
The notion of almost elementariness for a locally compact Hausdorff étale groupoid G with a compact unit space was introduced by the authors as a sufficient condition ensuring the reduced groupoid C^*-algebra C^*r(G) is (tracially) 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 C of Elliott invariants and a C^*-algebra A with Ell(A)∈ 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 Z.