**Mathematical Surveys and Monographs**

Volume: 241;
2019;
398 pp;
Hardcover

MSC: Primary 46;
Secondary 22

**Print ISBN: 978-1-4704-5133-2
Product Code: SURV/241**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

**Electronic ISBN: 978-1-4704-5409-8
Product Code: SURV/241.E**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

#### You may also like

#### Supplemental Materials

# A Tool Kit for Groupoid \(C^{*}\)-Algebras

Share this page
*Dana P. Williams*

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.

#### Readership

Graduate students and researchers interested in \(C^{*}\)-algebras.

#### Reviews & Endorsements

The book is written as a textbook with exercises at the end of each chapter, which is ideal for experts, but for the rest of us, this is a superb reference for particular topics that are currently only to be found scattered throughout the literature.

-- Mark V. Lawson, Heriot-Watt University

This graduate-level textbook is a comprehensive,
readable introduction to the fundamental theory of groupoid
C*-algebras. No textbook can make groupoid C*-theory easy, but

-- Elizabeth Gillaspy, University of Montana

#### Table of Contents

# Table of Contents

## A Tool Kit for Groupoid $C^{*}$-Algebras

- Cover Cover11
- Title iii4
- Copyright iv5
- Contents vii8
- Introduction xi12
- Chapter 1. From Groupoid to Algebra 118
- Chapter 2. Groupoid Actions and Equivalence 2946
- 2.1. Sectionformat {Groupoid Actions}{1} 2946
- 2.2. Sectionformat {The Mackey-Glimm-Ramsay Dichotomy}{1} 3855
- 2.3. Sectionformat {Equivalence}{1} 4360
- 2.4. Sectionformat {Generalized Morphisms}{1} 5269
- 2.5. Sectionformat {Linking Groupoids}{1} 5673
- 2.6. Sectionformat {The Equivalence Theorem}{1} 5976
- 2.7. Sectionformat {Some Important Examples}{1} 6077
- 2.8. Sectionformat {Transitive Groupoid cs -Algebras}{1} 6279

- Chapter 3. Measure Theory 6986
- 3.1. Sectionformat {Radon Families}{1} 6986
- 3.2. Sectionformat {$pi $-systems}{1} 7289
- 3.3. Sectionformat {Complex Radon Measures}{1} 7996
- 3.4. Sectionformat {The Fell Topology on the Space of Subgroups}{1} 8198
- 3.5. Sectionformat {Borel Hilbert Bundles}{1} 85102
- 3.6. Sectionformat {The Hilbert Space $L^2(X,H )$}{1} 87104
- 3.7. Sectionformat {The Hilbert Space $ltpguh $}{1} 90107
- 3.8. Sectionformat {The Quotient Borel Structure}{1} 93110

- Chapter 4. Proof of the Equivalence Theorem 95112
- Chapter 5. Basic Representation Theory 107124
- 5.1. Sectionformat {Invariant Ideals}{1} 107124
- 5.2. Sectionformat {The Support of a Representation}{1} 111128
- 5.3. Sectionformat {Inducing Representations}{1} 113130
- 5.4. Sectionformat {Supports of Induced Representations}{1} 120137
- 5.5. Sectionformat {Irreducible Representations}{1} 123140
- 5.6. Sectionformat {cs -Bundles}{1} 125142
- 5.7. Sectionformat {Type Structure}{1} 128145
- 5.8. Sectionformat {Closed Orbits}{1} 129146
- 5.9. Sectionformat {Inducing Irreducible Representations}{1} 135152
- 5.10. Sectionformat {The Non-Smooth Case}{1} 139156

- Chapter 6. The Existence and Uniqueness of Haar Systems 141158
- 6.1. Sectionformat {First Steps}{1} 141158
- 6.2. Sectionformat {Group Bundles}{1} 143160
- 6.3. Sectionformat {The Isotropy Groupoid and the Isotropy Map}{1} 145162
- 6.4. Sectionformat {Haar Systems on 'Etale Groupoids}{1} 150167
- 6.5. Sectionformat {Haar Systems on Equivalent Groupoids}{1} 151168
- 6.6. Sectionformat {Some Applications}{1} 153170

- Chapter 7. Unitary Representations 157174
- Chapter 8. Renault's Disintegration Theorem 169186
- Chapter 9. Amenability for Groupoids 187204
- 9.1. Sectionformat {Some Comments on the Group Case}{1} 187204
- 9.2. Sectionformat {First Definitions}{1} 189206
- 9.3. Sectionformat {Amenable Groupoids}{1} 197214
- 9.4. Sectionformat {Amenable Maps}{1} 203220
- 9.5. Sectionformat {Amenability and Equivalence}{1} 216233
- 9.6. Sectionformat {Topological Invariant Means}{1} 218235
- 9.7. Sectionformat {Borel Equivalence}{1} 227244
- 9.8. Sectionformat {Applications and Examples}{1} 235252

- Chapter 10. Measurewise Amenability for Groupoids 243260
- 10.1. Sectionformat {Means}{1} 243260
- 10.2. Sectionformat {Measurewise Invariant Means}{1} 252269
- 10.3. Sectionformat {Fun with Means}{1} 255272
- 10.4. Sectionformat {Measurewise Amenability and Equivalence}{1} 266283
- 10.5. Sectionformat {Amenable Measures}{1} 274291
- 10.6. Sectionformat {An Application}{1} 276293

- Chapter 11. Comments on Simplicity 279296
- Appendix A. Duals and Topological Vector Spaces 315332
- A.1. Sectionformat {The Strict Dual}{1} 315332
- A.2. Sectionformat {Projective Tensor Products}{1} 317334
- A.3. Sectionformat {The Dual of $coxlone $}{1} 319336
- A.4. Sectionformat {The Dual of $liytlox $}{1} 320337
- A.5. Sectionformat {A Dense Subspace of $E ^{**}$}{1} 328345
- A.6. Sectionformat {An Alternative Topology}{1} 329346

- Appendix B. Remarks on Blanchard's Theorem 331348
- Appendix C. The Inductive Limit Topology 343360
- Appendix D. Ramsay Almost Everywhere 345362
- Answers to Some of the Exercises 353370
- Bibliography 387404
- Notation and Symbol Index 393410
- Index 395412
- Back cover Back cover1417