The York Research Database

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.