Natsume, Toshikazu; Nest, Ryszard; Peter, Ingo Strict quantizations of symplectic manifolds. (English) Zbl 1064.53062 Lett. Math. Phys. 66, No. 1-2, 73-89 (2003). 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. Reviewer: Marcela Popescu (Craiova) Cited in 11 Documents 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) Keywords:deformation quantization; noncommutative manifolds; strict quantization; symplectic manifold; exactness of a discrete group Citations:Zbl 0526.58023; Zbl 0719.58042; Zbl 0835.46057; Zbl 1015.46040; Zbl 0961.46042 PDFBibTeX XMLCite \textit{T. Natsume} et al., Lett. Math. Phys. 66, No. 1--2, 73--89 (2003; Zbl 1064.53062) Full Text: DOI