  We provide a deformation quantization, in the sense of Rieffel, for
\textit{all} globally hyperbolic spacetimes with a Poisson structure. The
Poisson structures have to satisfy Fedosov type requirements in order for the
deformed product to be associative. We apply the novel deformation to quantum
field theories and their respective states and we prove that the deformed state
(i.e.\ a state in non-commutative spacetime) has a singularity structure
resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is
Hadamard. This proves that the Hadamard condition, and hence the quantum field
theoretical implementation of the equivalence principle is a general concept
that holds in spacetimes with quantum features (i.e. a non-commutative
spacetime).
