# Groupoids in Analysis, Geometry, and Physics

Share this page *Edited by *
*Arlan Ramsay; Jean Renault*

Groupoids often occur when there is symmetry of a nature not expressible in
terms of groups. Other uses of groupoids can involve something of a dynamical
nature. Indeed, some of the main examples come from group actions. It should
also be noted that in many situations where groupoids have been used, the main
emphasis has not been on symmetry or dynamics issues. For example, a foliation
is an equivalence relation and has another groupoid associated with it, called
the holonomy groupoid. While the implicit symmetry and dynamics are relevant,
the groupoid records mostly the structure of the space of leaves and the
holonomy. More generally, the use of groupoids is very much related to various
notions of orbit equivalence. The point of view that groupoids describe
“singular spaces” can be found in the work of A. Grothendieck and
is prevalent in the non-commutative geometry of A. Connes.

This book presents the proceedings from the Joint Summer Research Conference
on “Groupoids in Analysis, Geometry, and Physics” held in Boulder,
CO. The book begins with an introduction to ways in which groupoids allow a
more comprehensive view of symmetry than is seen via groups. Topics range from
foliations, pseudo-differential operators, \(KK\)-theory, amenability,
Fell bundles, and index theory to quantization of Poisson manifolds. Readers
will find examples of important tools for working with groupoids.

This book is geared to students and researchers. It is intended to improve
their understanding of groupoids and to encourage them to look further while
learning about the tools used.

# Table of Contents

## Groupoids in Analysis, Geometry, and Physics

- Contents vii8 free
- Introduction ix10 free
- Groupoids: Unifying internal and external symmetry. A tour through some examples 114 free
- A primer for the Brauer group of a groupoid 2134
- Amenable groupoids 3548
- The role of groupoids in classification theory: A new approach. The UHF algebra case 4760
- Bundles over groupoids 6780
- Groupoids and foliations 8396
- Etale groupoids, derived categories, and operations 101114
- The analytic index for proper, Lie groupoid actions 115128
- Groupoid C*-algebras and operator K-theory 137150
- Groupoids of manifolds with corners and index theory 147160
- Quantization of Poisson algebras associated to Lie algebroids 159172
- 1. Introduction 159172
- 2. Lie groupoids and Lie algebroids 163176
- 3. The C*-algebra of a Lie groupoid 166179
- 4. Strict deformation quantization 169182
- 5. Continuous fields of groupoids 172185
- 6. The tangent groupoid 174187
- 7. The Fourier transform on vector bundles 177190
- 8. Weyl quantization 180193
- 9. The local structure of the Poisson bracket 182195
- 10. Proof of Dirac's condition 185198
- 11. Examples and comments 187200
- References 190203