Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras

Abstract

In this paper, we study the ideal structure of reduced $C^{\ast }$-algebras $C_r^{\ast }(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^{\ast }(G)$ 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 $C_r^{\ast }(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^{\ast }(G)$. We also establish a sufficient condition on the ample groupoid $G$ that ensures pure infiniteness of $C_r^{\ast }(G)$ 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 $C_r^{\ast }(G)$ is a simple $C^{\ast }$-algebra which is either stably finite or strongly purely infinite.

25 pages To appear in Ergodic Theory and Dynamical Systems

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• Going-Down functors and the Künneth formula for crossed products by étale groupoids
• The type semigroup, comparison and almost finiteness for ample groupoids
• A dichotomy for groupoid C*-algebras
• Amenability and paradoxicality in semigroups and C*-algebras
• Quasi-local Algebras and Asymptotic Expanders
• A generalized type semigroup and dynamical comparison
• The Furstenberg Boundary of a Groupoid
• Classification of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Some results regarding the ideal structure of C*-algebras of étale groupoids
• The ideal intersection property for essential groupoid C*-algebras
• Purely infinite locally compact Hausdorff étale groupoids and their $C^{\ast }$-algebras
• Strict comparison for $C^{\ast }$-algebras arising from Almost finite groupoids
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• Algebraic actions I. C*-algebras and groupoids
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• Pure infiniteness and paradoxicality for graph $C^{\ast }$-algebras
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• Dense subalgebras of purely infinite simple groupoid C*-algebras
• Representation theory of topological full groups of étale groupoids and paradoxicality