Tannaka Duality for Proper Lie Groupoids

Classical Tannaka-Krein duality theory leads to the result that a compact group can be reconstructed form the algebra of its representative functions. It is natural to ask for a generalization of the aforesaid duality theory to the realm of Lie groupoids, where proper groupoids would play the same role as compact groups. By replacing the category of smooth vector bundles over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.