Quantization of Poisson Algebras Associated To

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers A0 = C(G) ≃ C0(G) and A~ = C (G) for ~ 6= 0. The same is true for the corresponding reduced C-algebras. Our results have applications to, e.g., transformation group C-algebras, K-theory, and index theory.