Deformation quantization using groupoids. The case of toric manifolds

In the framework of C∗-algebraic deformation quantization we propose a notion of a deformation groupoid which could apply to known examples such as Connes’s tangent groupoid of a manifold, its generalization by Landsman and Ramazan, Rieffel’s noncommutative torus, and even Landi’s noncommutative 4-sphere. We construct such a groupoid for a wide class of TnTn-spaces, that generalizes the one given for CnCn by Bellissard and Vittot. In particular, using the geometric properties of the moment map discovered in the 1980’s by Atiyah, Delzant, Guillemin and Sternberg, it provides a C∗-algebraic deformation quantization for all toric manifolds, including the 2-sphere and all complex projective spaces.