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Deformation Quantization for Actions of $$R^d$$
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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

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.

• 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