Omori, Hideki; Maeda, Yoshiaki; Yoshioka, Akira Existence of a closed star product. (English) Zbl 0771.58017 Lett. Math. Phys. 26, No. 4, 285-294 (1992). After introducing the notion of a strongly closed star product A. Connes, M. Flato and D. Sternheimer [Lett. Math. Phys. 24, 1-12 (1992; Zbl 0767.55005)] placed a question mark against the existence of such a product for general symplectic manifolds.The authors give a definite affirmative answer by proving that on every symplectic manifold there exists a strongly closed star product. This they do by showing that on any paracompact \(C^ \infty\) symplectic manifold there exists a real Weyl manifold. For real Weyl manifolds there exists an involutive anti automorphism on the manifold: by using this the authors give a complexification relating to a quaternion ring. Reviewer: C.S.Sharma (London) Cited in 1 ReviewCited in 10 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 46L85 Noncommutative topology 46L87 Noncommutative differential geometry 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics Keywords:closed star product; Weyl diffeomorphism; symplectic manifold; deformation of the Poisson algebra Citations:Zbl 0767.55005 PDF BibTeX XML Cite \textit{H. Omori} et al., Lett. Math. Phys. 26, No. 4, 285--294 (1992; Zbl 0771.58017) Full Text: DOI OpenURL References: [1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowiz, A., and Sternheimer, D., Deformation theory and quantization I, Ann. Phys. 111, 61-110 (1978). · Zbl 0377.53024 [2] Connes, A., Flato, M. and Sternheimer, D., Closed star products and cyclic cohomology, Lett. Math. Phys. 24, 1-12 (1992). · Zbl 0767.55005 [3] Karasev, M. V. and Nazaikinskii, V. E., On the quantization of rapidly oscillating symbols, Math. U.S.S.R. Izv. 34, 737-764 (1978). · Zbl 0391.47036 [4] Moyal, J. E., Qunatum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45, 99-124 (1949). · Zbl 0031.33601 [5] Omori, H., Infinite Dimensional Lie Groups (in Japanese), Kinokuni-ya, 1978. · Zbl 0573.58005 [6] Omori, H., Maeda, Y., and Yoshioka, A., Weyl manifolds and deformation quantization, Adv. in Math. 85, 224-255 (1991). · Zbl 0734.58011 [7] DeWilde, M. and Lecomte, P. B., Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7, 487-496 (1983). · Zbl 0526.58023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.