2017/07/31 by Christian Bönicke, Kang Li · 44 citations
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Combinatorics #Computer science #Discrete mathematics #Double groupoid #Ideal (ethics) #Invariant (physics) #Mathematics #Projection (relational algebra) #Pure mathematics #Semigroup #Space (punctuation) #Unit (ring theory) #Zero (linguistics) #math.OA
paper · pdf · open access · doi:10.1017/etds.2018.39
published in Ergodic Theory and Dynamical Systems 40(1), 34-63 (Cambridge University Press) · 25 pages To appear in Ergodic Theory and Dynamical Systems
openalex created_date 2017/07/21 · arxiv created 2018/04/26 · openalex publication_date 2018/06/14 · arxiv updated 2019/01/29 · openalex updated_date 2026/08/05
In this paper, we study the ideal structure of reduced C∗ -algebras Cr∗ (G) associated to étale groupoids G . In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in Cr∗ (G) and the open invariant subsets of the unit space G(0) of G . As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then Cr∗ (G) is (strongly) purely infinite if and only if every non-zero projection in C0(G(0)) is properly infinite in Cr∗ (G) . We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of Cr∗ (G) in terms of paradoxicality of compact open subsets of the unit space G(0) . Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: let G be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then Cr∗ (G) is a simple C∗ -algebra which is either stably finite or strongly purely infinite.