Equivariant Kasparov theory and groupoids. I (Q1288993)

The content of the paper is part of the author's thesis. The main results have been announced [\textit{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 \textit{M. Hilsum} and \textit{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.