Ideal structure and pure infiniteness of ample groupoid C∗-algebras
Boenicke C, Li K (2020)
Publication Type: Journal article
Publication year: 2020
Journal
Book Volume: 40
Pages Range: 34-63
Journal Issue: 1
DOI: 10.1017/etds.2018.39
Abstract
In this paper, we study the ideal structure of reduced -algebras associated to étale groupoids . In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in and the open invariant subsets of the unit space of . As a consequence, we show that if is an inner exact, essentially principal, ample groupoid, then is (strongly purely infinite if and only if every non-zero projection in is properly infinite in . We also establish a sufficient condition on the ample groupoid that ensures pure infiniteness of in terms of paradoxicality of compact open subsets of the unit space . Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: let be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then is a simple -algebra which is either stably finite or strongly purely infinite.
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Germany (DE)How to cite
APA:
Boenicke, C., & Li, K. (2020). Ideal structure and pure infiniteness of ample groupoid C∗-algebras. Ergodic Theory and Dynamical Systems, 40(1), 34-63. https://dx.doi.org/10.1017/etds.2018.39
MLA:
Boenicke, Christian, and Kang Li. "Ideal structure and pure infiniteness of ample groupoid C∗-algebras." Ergodic Theory and Dynamical Systems 40.1 (2020): 34-63.
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