Ideal structure and pure infiniteness of ample groupoid C* -algebras

Bönicke, C. and Li, K. (2020) Ideal structure and pure infiniteness of ample groupoid C* -algebras. Ergodic Theory and Dynamical Systems, 40(1), pp. 34-63. (doi: 10.1017/etds.2018.39)

Abstract

In this paper, we study the ideal structure of reduced C* -algebras C*, (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 C*, (G) and the open invariant subsets of the unit space G(o) of G. As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then C*, (G) is (strongly) purely infinite if and only if every non-zero projection in Co (G(o)) is properly infinite in C*, (G). We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of C*, (G) in terms of paradoxicality of compact open subsets of the unit space G(o). 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 C*, (G) is a simple C* -algebra which is either stably finite or strongly purely infinite.

• Item Type: Articles
• Additional Information: The first author is supported by Deutsche Forschungsgemeinschaft (SFB 878). The second author is supported by the Danish Council for Independent Research (DFF-5051- 00037) and partially supported by the DFG (SFB 878).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bonicke, Dr Christian
• Authors: Bönicke, C., and Li, K.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Ergodic Theory and Dynamical Systems
• Publisher: Cambridge University Press
• ISSN: 0143-3857
• ISSN (Online): 1469-4417
• Published Online: 14 June 2018