Abstract
A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.
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Landsman, N.P. The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart. Letters in Mathematical Physics 54, 43–59 (2000). https://doi.org/10.1023/A:1007669418336
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DOI: https://doi.org/10.1023/A:1007669418336