Natural star products on symplectic manifolds and quantum moment maps

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We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.