Quantum orbit method for the Connes-Landi-Matsumoto 3-sphere

We discuss the correspondence between the symplectic foliation of a Poisson structure on the 3-sphere and the unitary spectrum of its C∗-algebraic quantization, known as Connes-Landi-Matsumoto 3-sphere. Quantization is obtained via symplectic groupoid quantization and this allows to understand various peculiarities of such correspondence. In the last section we discuss how this relates to quantization of Dirac structures (and foliations) and speculate on how to extend thiscorrespondence to general locally abelian Poisson manifolds.