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Deformation Quantization for Actions of \(R^d\)
Marc A. Rieffel

Memoirs of the American Mathematical Society
1993; 93 pp; softcover
Volume: 106
ISBN-10: 0-8218-2575-5
ISBN-13: 978-0-8218-2575-4
List Price: US$36
Individual Members: US$21.60
Institutional Members: US$28.80
Order Code: MEMO/106/506
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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.


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

Table of Contents

  • Oscillatory integrals
  • The deformed product
  • Function algebras
  • The algebra of bounded operators
  • Functoriality for the operator norm
  • Norms of deformed deformations
  • Smooth vectors, and exactness
  • Continuous fields
  • Strict deformation quantization
  • Old examples
  • The quantum Euclidean closed disk and quantum quadrant
  • The algebraists quantum plane, and quantum groups
  • References
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