## 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

Zbl 0767.55005
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### References:

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