**Memoirs of the American Mathematical Society**

1993;
93 pp;
Softcover

MSC: Primary 46;
Secondary 35

Print ISBN: 978-0-8218-2575-4

Product Code: MEMO/106/506

List Price: $36.00

Individual Member Price: $21.60

**Electronic ISBN: 978-1-4704-0083-5
Product Code: MEMO/106/506.E**

List Price: $36.00

Individual Member Price: $21.60

# Deformation Quantization for Actions of $R^{d}$

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*Marc A. Rieffel*

This work describes a general construction of a deformation quantization for any Poisson bracket on a manifold which comes from an action of \(R^d\) on that manifold. These deformation quantizations are strict, in the sense that the deformed product of any two functions is again a function and that there are corresponding involutions and operator norms. Many of the techniques involved are adapted from the theory of pseudo-differential operators. The construction is shown to have many favorable properties. A number of specific examples are described, ranging from basic ones such as quantum disks, quantum tori, and quantum spheres, to aspects of quantum groups.

#### Table of Contents

# Table of Contents

## Deformation Quantization for Actions of $R^{d}$

- Contents v6 free
- Introduction vii8 free
- Chapter 1. Oscillatory Integrals 112 free
- Chapter 2. The Deformed Product 1122
- Chapter 3. Function Algebras 2334
- Chapter 4. The Algebra of Bounded Operators 2940
- Chapter 5. Functoriality for the Operator Norm 4051
- Chapter 6. Norms of Deformed Deformations 4556
- Chapter 7. Smooth Vectors, and Exactness 5061
- Chapter 8. Continuous Fields 5364
- Chapter 9. Strict Deformation Quantization 6172
- Chapter 10. Old Examples 6980
- Chapter 11. The Quantum Euclidean Closed Disk and Quantum Quadrant 7384
- Chapter 12. The Algebraists Quantum Plane, and Quantum Groups 8293
- References 89100

#### Readership

Researchers and advanced graduate students, especially those working in quantum geometry.