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**Existence of a closed star product.**
*(English)*
Zbl 0771.58017

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.

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)

### 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
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\textit{H. Omori} et al., Lett. Math. Phys. 26, No. 4, 285--294 (1992; Zbl 0771.58017)

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### 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 |

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