Alternative commutation relations, star products and tomography

Invertible maps from operators of quantum observables onto functions of c-number arguments and their associative products are first assessed. Different types of maps such as the Weyl-Wigner-Stratonovich map and s-ordered quasi-distribution are discussed. The recently introduced symplectic tomography map of observables (tomograms) related to the Heisenberg-Weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c-number functions. The star product for symbols of the quantum observable for each one of the maps (including the tomographic map) and explicit relations among different star products are obtained. Deformations of the Moyal star product and alternative commutation relations are also considered.