2001/07/23 by N. P. Landsman, Landsman, N. P.
Mathematics · Physics and Astronomy · #46L65 #53D20 #53D55 #81S10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Symplectic Geometry (math.SG) #math-ph #math.MP #math.OA #math.SG #msc:46L65 #msc:53D20 #msc:53D55 #msc:81S10
paper · pdf · doi:10.48550/arxiv.math-ph/0107023
20 pages LaTeX. Major revision
arxiv created 2002/02/13 · arxiv updated 2009/11/30
Notwithstanding known obstructions to this idea, we formulate an attempt to turn quantization into a functorial procedure. We define a category PO of Poisson manifolds, whose objects are integrable Poisson manifolds and whose arrows are isomorphism classes of regular Weinstein dual pairs; it follows that identity arrows are symplectic groupoids, and that two objects are isomorphic in PO iff they are Morita equivalent in the sense of P. Xu. It has a subcategory LPO that has duals of integrable Lie algebroids as objects and cotangent bundles as arrows. We argue that naive C*-algebraic quantization should be functorial from LPO to the well-known category KK, whose objects are separable C*-algebras and whose arrows are Kasparov's KK-groups. This limited functoriality of quantization would already imply the Atiyah-Singer index theorem, as well as its far-reaching generalizations developed by Connes and others. In the category KK, isomorphism of objects implies isomorphism of K-theory groups, so that the functoriality of quantization on all of PO would imply that Morita equivalent Poisson algebras are quantized by C*-algebras with isomorphic K-theories. Finally, we argue that the correct codomain for the possible functoriality of quantization is the category RKK(I), which takes the deformation aspect of quantization into account.