On local equivalence of star-products on Poisson manifolds

We present a proof that every star-product defined on a Poisson manifold and written in a given quantum canonical coordinate system is uniquely equivalent with a Moyal product associated with this coordinate system. The equivalence is assumed to satisfy some additional conditions which guarantee its uniqueness. Moreover, the systematic construction of such equivalence is presented and a formula for this equivalence in a case of a particular class of star-products is given, to the fourth order in $\hbar$.