**Contemporary Mathematics**

Volume: 449;
2008;
446 pp;
Softcover

MSC: Primary 22; 28; 43; 46; 58; 81;

Print ISBN: 978-0-8218-4225-6

Product Code: CONM/449

List Price: $123.00

Individual Member Price: $98.40

**Electronic ISBN: 978-0-8218-8128-6
Product Code: CONM/449.E**

List Price: $123.00

Individual Member Price: $98.40

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# Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey

Share this page *Edited by *
*Robert S. Doran; Calvin C. Moore; Robert J. Zimmer*

George Mackey was an extraordinary mathematician of great power and vision. His profound contributions to representation theory, harmonic analysis, ergodic theory, and mathematical physics left a rich legacy for researchers that continues today. This book is based on lectures presented at an AMS special session held in January 2007 in New Orleans dedicated to his memory. The papers, written especially for this volume by internationally-known mathematicians and mathematical physicists, range from expository and historical surveys to original high-level research articles. The influence of Mackey's fundamental ideas is apparent throughout. The introductory article contains recollections from former students, friends, colleagues, and family as well as a biography describing his distinguished career as a mathematician at Harvard, where he held the Landon D. Clay Professorship of Mathematics.

Topics examined here include recent results on induced representations, virtual groups, the Mackey Machine and crossed products, representations of Baumslag-Solitar groups, the Radon transform and the heat equation, groupoids in the study of wavelets, and quantum theory. The in-depth historical surveys of Mackey's work on representation theory, ergodic theory, and physics, together with recent developments inspired by his fundamental work will be of considerable interest to both graduate students and researchers alike.

#### Table of Contents

# Table of Contents

## Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey

- Contents vii8 free
- Preface ix10 free
- George Mackey 1916–2006 112 free
- Decay to Zero of Matrix Coefficients at Adjoint Infinity 4354
- Induced Representations, Intertwining Operators and Transfer 5162
- MASA's and Certain Type I Closed Faces of C*-algebras 6980
- A Duality Approach to Representations of Baumslag-Solitar Groups 99110
- 1. Introduction 100111
- 2. Wavelet Sets 102113
- 3. Spectral Types 102113
- 4. IndGB(X) 105116
- 5. Haar Measure 110121
- 6. The Plancherel Formula for G 114125
- 7. Finite Dimensional Representations 118129
- 8. Dilations 121132
- 9. Wavelet Questions and Answers for the Group G = B xα Z 122133
- 10. Concluding Remarks 124135

- The Mackey Machine for Crossed Products: Inducing Primitive Ideals 129140
- Classifying the Unclassifiables 137148
- The Mackey Analogy and K-Theory 149160
- Some Recent Applications of Induced Representations 173184
- Groupoid Methods in Wavelet Analysis 193204
- Quantum Theory and Relativity 209220
- Thoughts about George Mackey and his lmprimitivity Theorem 247258
- Virtual Groups 45 Years Later 263274
- Spherical Characters: The Supercuspidal Case 301312
- Representation Theory, Radon Transform and the Heat Equation on a Riemannian Symmetric Space 315326
- Projective Representations and the Mackey Obstruction – A Survey 345356
- Virtual Groups for Group Representations 379390
- A Global View of Equivariant Vector Bundles and Dirac Operators on Some Compact Homogeneous Spaces 399410
- George Mackey and His Work on Representation Theory and Foundations of Physics 417428
- 1. Introduction 418429
- 2. Stone-Von Neumann Theorem, Systems of Imprimitivity, and the Imprimitivity Theorem 419430
- 3. Semi Direct Products and the Little Group Method. Representations of Finite Length and Orbit Schemes 424435
- 4. Super Lie Groups and their Systems of Imprimitivity 427438
- 5. Intertwining Numbers, Double Coset Decompositions, and Irreducibility of Induced Representations 429440
- 6. Projective Representations, Borel Cohomology of Groups, and the Metaplectic Representation 431442
- 7. Foundations of Physics: Hidden Variables and the Mackey-Gleason Theorem 437448

#### Readership

Graduate students and research mathematicians interested in ergodic theory and mathematicial physics.