{"title":"Almost elementary groupoid models for $C^*$-algebras","authors":"Xin Ma, Jianchao Wu","doi":"arxiv-2407.05251","DOIUrl":null,"url":null,"abstract":"The notion of almost elementariness for a locally compact Hausdorff \\'{e}tale\ngroupoid $\\mathcal{G}$ with a compact unit space was introduced by the authors\nas a sufficient condition ensuring the reduced groupoid $C^*$-algebra\n$C^*_r(\\mathcal{G})$ is (tracially) $\\mathcal{Z}$-stable and thus classifiable\nunder additional natural assumption. In this paper, we explore the converse\ndirection and show that many groupoids in the literature serving as models for\nclassifiable $C^*$-algebras are almost elementary. In particular, for a large\nclass $\\mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with\n$\\operatorname{Ell}(A)\\in \\mathcal{C}$, we show that $A$ is classifiable if and\nonly if $A$ possesses a minimal, effective, amenable, second countable, almost\nelementary groupoid model, which leads to a groupoid-theoretic characterization\nof classifiability of $C^*$-algebras with certain Elliott invariants. Moreover,\nwe build a connection between almost elementariness and pure infiniteness for\ngroupoids and study obstructions to obtaining a transformation groupoid model\nfor the Jiang-Su algebra $\\mathcal{Z}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.05251","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $\mathcal{Z}$.