The ⋆-value equation and Wigner distributions in noncommutative Heisenberg algebras

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.Dedicated to Mike Ryan on his sixtieth birthday, who as a scientist always understood that it is nice to be good, but that it is better to be nice.