Phase space quantization and the operator moment problem

We consider questions related to a quantization scheme in which a classical variable f:Ω→ ℝ on a phase space Ω is associated with a (preferably unique) semispectral measure E^f, such that the moment operators of E^f are required to be of the form Λ(f ^k), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally ℝ², where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map F is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.