Étale groupoids and their $C^{\ast }$-algebras

Abstract

These notes were written as supplementary material for a five-hour lecture series presented at the Centre de Recerca Mathemàtica at the Universitat Autònoma de Barcelona from the 13th to the 17th of March 2017. The intention of these notes is to give a brief overview of some key topics in the area of $C^{\ast }$-algebras associated to étale groupoids. The scope has been deliberately contained to the case of étale groupoids with the intention that much of the representation-theoretic technology and measure-theoretic analysis required to handle general groupoids can be suppressed in this simpler setting. A published version of these notes will appear in the volume tentatively titled "Operator algebras and dynamics: groupoids, crossed products and Rokhlin dimension" by Gabor Szabo, Dana P. Williams and myself, and edited by Francesc Perera, in the series "Advanced Courses in Mathematics. CRM Barcelona." The pagination of this arXiv version is not identical to Birkhäuser's style, but I have tried to make it close. The theorem numbering should be correct. I'm grateful to the CRM and Birkhäuser for allowing me to post a version on arXiv.

57+iv pages; v2: a number of typos fixed, and the statement of Theorem 5.2.7 corrected

References in corpus (2)

• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras

Cited by in corpus (20)

• Reconstruction of groupoids and C*-rigidity of dynamical systems
• General Non-Commutative Locally Compact Locally Hausdorff Stone Duality
• Graded C*-algebras and twisted groupoid C*-algebras
• An Algebraic Approach to the Weyl Groupoid
• Twisted Steinberg algebras
• C*-algebras, groupoids and covers of shift spaces
• $C^{\ast }$-simplicity and representations of topological full groups of groupoids
• Almost elementary étale groupoids
• The Furstenberg Boundary of a Groupoid
• Purely infinite locally compact Hausdorff étale groupoids and their $C^{\ast }$-algebras
• Quotients of étale groupoids and the abelianizations of groupoid C*-algebras
• Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability
• Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras
• Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States
• Honours thesis: Exact sequences in graded $KK$-theory for Cuntz-Pimsner algebras
• Limit operator theory for groupoids
• Invariant sets and normal subgroupoids of universal étale groupoids induced by congruences of inverse semigroups
• Sturmian subshifts and their C*-algebras
• Comparison and Simplicity of Commutator Subgroups of Full Groups
• $C^{\ast }$-uniqueness Results for Groupoids