A deformation quantization theory for noncommutative quantum mechanics

We show that the deformation quantization of noncommutative quantum mechanics previously considered by Dias and Prata ["Weyl-Wigner formulation of noncommutative quantum mechanics," J. Math. Phys. 49, 072101 (2008)] and Bastos, Dias, and Prata ["Wigner measures in non-commutative quantum mechanics," e-print arXiv:math-ph/0907.4438v1; Commun. Math. Phys. (to appear)] can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef ["A new approach to the star-genvalue equation," Lett. Math. Phys. 85, 173-183 (2008)]. (C) 2010 American Institute of Physics. [doi:10.1063/1.3436581]