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Strict quantizations of symplectic manifolds. (English) Zbl 1064.53062

The deformation quantization of a symplectic manifold was proven by M. De Wilde and P. B. A. Lecomte, [Lett. Math. Phys. 7, 487–496 (1983; Zbl 0526.58023)]. The subject of the noncommutative differential geometry imposed the problem of a strict quantization [A. J.-L. Sheu, Commun. Math. Phys. 135, No. 2, 217–232 (1991; Zbl 0719.58042)] of a Poisson manifold. The noncommutative \(2\)-torus is a strict quantization of the \(2\)-torus with the canonical symplectic structure. The main result of the paper extends this situation, in the following theorem:
Let \(M\) be a closed symplectic manifold such that \(\pi _{1}(M)\) is exact and \(\pi _{2}(M)=0\). Then \(M\) has a strict quantization. The notion of exactness of a discrete group has been studied in [Math. Ann. 303, No. 4, 677–697 (1995; Zbl 0835.46057)] by E. Kirchberg and S. Wassermann. As the authors remark, it is possible that the topological conditions are redundant. The local method used in the paper is different from that used by the first two authors for Riemannian surfaces of genus grater that one [Cuntz, Joachim (ed.) et al., \(C^*\)-algebras. Proceedings of the SFB-workshop, Münster, Germany, March 8-12, 1999. Berlin: Springer. 142–150 (2000; Zbl 1015.46040) and Commun. Math. Phys. 202, No. 1, 65–87 (1999; Zbl 0961.46042)], a situation covered also by the main result.

MSC:

53D55 Deformation quantization, star products
46L85 Noncommutative topology
32G10 Deformations of submanifolds and subspaces
81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization
58B34 Noncommutative geometry (à la Connes)
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