A groupoid approach to quantization

Many interesting C * -algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C * -algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the C * -algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid C * -algebras as quantizations of bundle affine Poisson structures.