Lie Groupoid C∗-Algebras and Weyl Quantization

A strict quantization of a Poisson manifold P on a subset I ⊆ R containing 0 as an accumulation point is defined as a continuous field of C∗-algebras {Ah̄}h̄∈I , with A0 = C0(P ), a dense subalgebra Ã0 of C0(P ) on which the Poisson bracket is defined, and a set of continuous cross-sections {Q(f )} f∈Ã0 for which Q0(f ) = f . Here Qh̄(f ∗) = Qh̄(f )∗ for all h̄ ∈ I , whereas for h̄ → 0 one requires that i[Qh̄(f ),Qh̄(g)]/h̄→ Qh̄({f, g}) in norm. For any Lie groupoid G, the vector bundle G∗ dual to the associated Lie algebroid G is canonically a Poisson manifold. Let A0 = C0(G∗), and for h̄ 6= 0 let Ah̄ = C∗(G) be theC∗-algebra of G. The family ofC∗-algebras {Ah̄}h̄∈[0,1] forms a continuous field, and we construct a dense subalgebra Ã0 ⊂ C0(G∗) and an associated family {Qh̄ (f )} of continuous cross-sections of this field, generalizing Weyl quantization, which define a strict quantization of G∗. Many known strict quantizations are a special case of this procedure. On P = T ∗Rn the maps Qh̄ (f ) reduce to standard Weyl quantization; for P = T ∗Q, where Q is a Riemannian manifold, one recovers Connes’ tangent groupoid as well as a recent generalization ofWeyl’s prescription.When G is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang–Mills field. In case that G is a Lie group (with Lie algebra g) one recovers Rieffel’s quantization of the Lie–Poisson structure on g∗. A transformation group C∗-algebra defined by a smooth action of a Lie group on a manifold Q turns out to be the quantization of the Poisson manifold g∗ ×Q defined by this action.