Deformation quantization for actions of \(\mathbb{R}^ d\). (English) Zbl 0798.46053

Mem. Am. Math. Soc. 506, 93 p. (1993).
Author’s summary: “Let \(\alpha\) be an action of \(\mathbb{R}^ d\) on a manifold \(M\). For any skew-symmetric operator \(J\) on \(\mathbb{R}^ d\) there is a corresponding Poisson bracket on \(M\) determined by \(\alpha\). We show how to construct a strict deformation quantization of the resulting Poisson manifold on which the deformed product of functions is again a function (not just a formal power series). More generally, if \(\alpha\) is an action of \(\mathbb{R}^ d\) on a \(C^*\)-algebra \(A\) (or, to a substantial extent, on a Fréchet algebra), we show how to construct a deformation quantization of \(A\). This construction has many favorable properties. Notably, we show that \(\alpha\)-equivariant short exact sequences of \(C^*\)-algebras are carried to short exact sequences. Many of the techniques which we employ are adapted from the theory of pseudo- differential operators.
We give a number of specific examples, such as quantum tori, quantum disks, and quantum quadrants. Notable are a quantization of a Poisson bracket on the plane which is closely related to certain quantum planes recently studied by algebraists, and some solutions with quantum SU(2).”
The content of this memories is as follows.
Chapter 1. Oscillatory integrals;
Chapter 2. The deformed product;
Chapter 3. Function algebras;
Chapter 4. The algebra of bounded operators;
Chapter 5. Functoriality for the operator norm;
Chapter 6. Norms of deformed deformations;
Chapter 7. Smooth vectors and exactness;
Chapter 8. Continuous fields;
Chapter 9. Strict deformation quantization;
Chapter 10. Old examples;
Chapter 11. The quantum Euclidean closed disk and quantum quadrants;
Chapter 12. The algebraists’ quantum plane and quantum group.
There are 90 references.


46L87 Noncommutative differential geometry
35S05 Pseudodifferential operators as generalizations of partial differential operators
46L60 Applications of selfadjoint operator algebras to physics
81S10 Geometry and quantization, symplectic methods
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