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Equivariant Kasparov theory and groupoids. I. (Théorie de Kasparov équivariante et groupoïdes. I.) (French) Zbl 0932.19004

The content of the paper is part of the author’s thesis. The main results have been announced [P.-Y. Le Gall, C. R. Acad. Sci., Paris, Sér. I 324, No. 6, 695-698 (1997; Zbl 0883.46041)]. Motivated by the work of M. Hilsum and G. Skandalis [Ann. Sci. Éc. Norm. Supér., Sér. IV 20, 325-390 (1987; Zbl 0656.57015)] on foliations and aiming at proofs of the Novikov and of the Baum-Connes conjectures for foliations the author extends Kasparov’s equivariant \(KK\)-theory from actions of groups to actions of groupoids. He establishes the existence of the product and of functoriality properties with respect to morphisms of groupoids.

MSC:

19K35 Kasparov theory (\(KK\)-theory)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
22A22 Topological groupoids (including differentiable and Lie groupoids)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
58J22 Exotic index theories on manifolds
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