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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.

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

Citations:

Zbl 0767.55005
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References:

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[2] Connes, A., Flato, M. and Sternheimer, D., Closed star products and cyclic cohomology, Lett. Math. Phys. 24, 1-12 (1992). · Zbl 0767.55005
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