## A groupoid approach to quantization.(English)Zbl 1154.46041

Summary: 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.

### MSC:

 46L65 Quantizations, deformations for selfadjoint operator algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids 22A22 Topological groupoids (including differentiable and Lie groupoids) 53D50 Geometric quantization 58A05 Differentiable manifolds, foundations
Full Text: