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Journal of Operator Theory

Volume 83, Issue 1, Winter 2020  pp. 95-138.

Simplicity criteria for \'etale groupoid $C^*$-algebras

Authors:  Danny Crytser (1), Gabriel Nagy (2)
Author institution:(1) Department of Mathematics, St. Lawrence University, 23 Romoda Drive, Canton, NY 13617, U.S.A.
(2) Department of Mathematics, Kansas State University, 1228 N. 17th Street, Manhattan, KS 66506, U.S.A.

Summary: We develop a framework suitable for obtaining simplicity criteria for reduced $C^*$-algebras of Hausdorff \'etale groupoids. This is based on the study of certain non-degenerate $C^*$-subalgebras (in the case of groupoids, the $C^*$-algebra of the interior isotropy bundle), for which one can control (non-unique) state extensions to the ambient $C^*$-algebra. As an application, we give simplicity criteria for reduced crossed products $C_0(Q)\rtimes_\mathrm{red} G$ by discrete groups.

Keywords: \'etale groupoids, groupoid $C^*$-algebras, simple $C^*$-algebras, essential inclusions, regular inclusions, crossed products

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