The Furstenberg Boundary of a Groupoid

2019/04/22 by Clemens Borys, Borys, Clemens
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1904.10062

openalex publication_date 2019/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the Furstenberg boundary of a locally compact Hausdorff étale groupoid, generalising the Furstenberg boundary for discrete groups, by providing a construction of a groupoid-equivariant injective envelope. Using this injective envelope, we establish the absence of recurrent amenable subgroups in the isotropy as a sufficient criterion for the intersection property of a locally compact Hausdorff étale groupoid with compact unit space and no fixed points. This yields a criterion for C*-simplicity of minimal groupoids.

Citations

• Uniformly recurrent subgroups and the ideal structure of reduced crossed products
• Étale groupoids and their C^*-algebras

Related

• C^*-simplicity and Ozawa conjecture for groupoid C^*-algebras, part I: injective envelopes
• Exact groupoids
• The ideal intersection property for essential groupoid C*-algebras
• Purely infinite locally compact Hausdorff étale groupoids and their C^*-algebras
• Furstenberg boundary of minimal actions