  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.
