Functoriality and Morita Equivalence of C * -algebras and Poisson Manifolds Associated to Lie Groupoids
نویسنده
چکیده
It is well known that a Lie groupoid G canonically defines both a C *-algebra C * (G) and a Poisson manifold A * (G). We show that the maps G → C * (G) and G → A * (G) are functorial with respect to suitably defined categories. The arrows (Hom-spaces) between Lie groupoids are taken to be isomorphism classes of regular bibundles (Hilsum–Skandalis maps), composed by a canonical bibundle tensor product. The appropriate arrows between C *-algebras are unitary isomorphism classes of Hilbert bimodules, composed by Rieffel's tensor product. Finally, integrable Poisson manifolds are connected by isomorphism classes of regular symplectic bimodules (dual pairs), composed by symplectic reduction. These choices have the additional advantage that the notion of Morita equivalence (as defined by Muhly–Renault–Williams for groupoids, by Rieffel for C *-algebras, and by Xu for Poisson manifolds), turns out to coincide with iso-morphism of objects in the pertinent category. It then trivially follows from their functoriality that the maps G → C * (G) and G → A * (G) preserve Morita equivalence.
منابع مشابه
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