A Tool Kit for Groupoid C∗ -Algebras

The construction of a $C^{•}$-algebra from a locally compact groupoid is an important generalization of the group $C^{•}$-algebra construction and of the transformation group $C^{•}$-algebra construction. Since their introduction in 1980, groupoid $C^{•}$-algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the Baum-Connes conjecture, and noncommutative geometry. This book provides a detailed introduction to this vast subject and is suitable for graduate students or any researcher who wants to use groupoid $C^{•}$-algebras in their work. The main focus is to equip the reader with modern versions of the basic technical tools used in the subject, which will allow the reader to understand fundamental results and make contributions to various areas in the subject. Thus, in addition to covering the basic properties and construction of groupoid $C^{•}$-algebras, the focus is to give a modern treatment of some of the major developments in the subject in recent years, including the Equivalence Theorem and the Disintegration Theorem. Also covered are the complicated subjects of amenability of groupoids and simplicity results. The book is reasonably self-contained and accessible to graduate students with a good background in operator algebras. Cover Title Copyright Contents Introduction Chapter 1. From Groupoid to Algebra 1.1. Sectionformat {Preliminaries}{1} 1.2. Sectionformat {Getting Started}{1} 1.3. Sectionformat {Haar Systems}{1} 1.4. Sectionformat {Building cs -Algebras}{1} Chapter 2. Groupoid Actions and Equivalence 2.1. Sectionformat {Groupoid Actions}{1} 2.2. Sectionformat {The Mackey-Glimm-Ramsay Dichotomy}{1} 2.3. Sectionformat {Equivalence}{1} 2.4. Sectionformat {Generalized Morphisms}{1} 2.5. Sectionformat {Linking Groupoids}{1} 2.6. Sectionformat {The Equivalence Theorem}{1} 2.7. Sectionformat {Some Important Examples}{1} 2.8. Sectionformat {Transitive Groupoid cs -Algebras}{1} Chapter 3. Measure Theory 3.1. Sectionformat {Radon Families}{1} 3.2. Sectionformat {$pi $-systems}{1} 3.3. Sectionformat {Complex Radon Measures}{1} 3.4. Sectionformat {The Fell Topology on the Space of Subgroups}{1} 3.5. Sectionformat {Borel Hilbert Bundles}{1} 3.6. Sectionformat {The Hilbert Space $L^2(X,H )$}{1} 3.7. Sectionformat {The Hilbert Space $ltpguh $}{1} 3.8. Sectionformat {The Quotient Borel Structure}{1} Chapter 4. Proof of the Equivalence Theorem 4.1. Sectionformat {The cs -Algebra of the Linking Groupoid}{1} 4.2. Sectionformat {Approximate Units}{1} Chapter 5. Basic Representation Theory 5.1. Sectionformat {Invariant Ideals}{1} 5.2. Sectionformat {The Support of a Representation}{1} 5.3. Sectionformat {Inducing Representations}{1} 5.4. Sectionformat {Supports of Induced Representations}{1} 5.5. Sectionformat {Irreducible Representations}{1} 5.6. Sectionformat {cs -Bundles}{1} 5.7. Sectionformat {Type Structure}{1} 5.8. Sectionformat {Closed Orbits}{1} 5.9. Sectionformat {Inducing Irreducible Representations}{1} 5.10. Sectionformat {The Non-Smooth Case}{1} Chapter 6. The Existence and Uniqueness of Haar Systems 6.1. Sectionformat {First Steps}{1} 6.2. Sectionformat {Group Bundles}{1} 6.3. Sectionformat {The Isotropy Groupoid and the Isotropy Map}{1} 6.4. Sectionformat {Haar Systems on 'Etale Groupoids}{1} 6.5. Sectionformat {Haar Systems on Equivalent Groupoids}{1} 6.6. Sectionformat {Some Applications}{1} Chapter 7. Unitary Representations 7.1. Sectionformat {Quasi-invariant Measures}{1} 7.2. Sectionformat {Unitary Representations}{1} 7.3. Sectionformat {An Example: Regular Representations}{1} Chapter 8. Renault's Disintegration Theorem 8.1. Sectionformat {The Statement}{1} 8.2. Sectionformat {The Proof}{1} Chapter 9. Amenability for Groupoids 9.1. Sectionformat {Some Comments on the Group Case}{1} 9.2. Sectionformat {First Definitions}{1} 9.3. Sectionformat {Amenable Groupoids}{1} 9.4. Sectionformat {Amenable Maps}{1} 9.5. Sectionformat {Amenability and Equivalence}{1} 9.6. Sectionformat {Topological Invariant Means}{1} 9.7. Sectionformat {Borel Equivalence}{1} 9.8. Sectionformat {Applications and Examples}{1} Chapter 10. Measurewise Amenability for Groupoids 10.1. Sectionformat {Means}{1} 10.2. Sectionformat {Measurewise Invariant Means}{1} 10.3. Sectionformat {Fun with Means}{1} 10.4. Sectionformat {Measurewise Amenability and Equivalence}{1} 10.5. Sectionformat {Amenable Measures}{1} 10.6. Sectionformat {An Application}{1} Chapter 11. Comments on Simplicity 11.1. Sectionformat {The Auxillary Groupoids}{1} 11.2. Sectionformat {Restricting to the Isotropy}{1} 11.3. Sectionformat {Renault's Simplicity Result}{1} 11.4. Sectionformat {The Amenable Case}{1} Appendix A. Duals and Topological Vector Spaces A.1. Sectionformat {The Strict Dual}{1} A.2. Sectionformat {Projective Tensor Products}{1} A.3. Sectionformat {The Dual of $coxlone $}{1} A.4. Sectionformat {The Dual of $liytlox $}{1} A.5. Sectionformat {A Dense Subspace of $E ^{**}$}{1} A.6. Sectionformat {An Alternative Topology}{1} Appendix B. Remarks on Blanchard's Theorem Appendix C. The Inductive Limit Topology Appendix D. Ramsay Almost Everywhere Answers to Some of the Exercises Bibliography Notation and Symbol Index Index The construction of a C*-algebra from a locally compact groupoid is an important generalization of the group C*-algebra construction and of the transformation group C*-algebra construction. Since their introduction in 1980, groupoid C*-algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the Baum-Connes conjecture, and noncommutative geometry. This book provides a detailed introduction to this vast subject and is suitable for graduate students or any researcher who wants to use groupoid C*-algebras in their work. The main focus is to equip the reader with modern versions of the basic technical tools used in the subject, which will allow the reader to understand fundamental results and make contributions to various areas in the subject. Thus, in addition to covering the basic properties and construction of groupoid C*-algebras, the focus is to give a modern treatment of some of the major developments in the subject in recent years, including the Equivalence Theorem and the Disintegration Theorem. Also covered are the complicated subjects of amenability of groupoids and simplicity results -- Prové de l'editor