Wigner Quantization on the Circle and R

We construct a deformation quantization for two cases of configuration spaces: the multiplicative group of positive real numbers R+ and the circle S 1 . In these cases we define the momenta using the Fourier transform. Using the identification of symbols of quantum observables — real functions on the phase space — with classical observables, we introduce a non–commutative ⋆–product between a pair of them.