Semi-exactitude du bifoncteur de Kasparov pour les actions moyennables

Consider an equivariant extension of graded separable G-algebras which admits a completely linear positive, grading preserving cross section (not necessary equivariant) of norm 1. We denote (X,G) an amenable topological transformation group in the sense of Anantharaman-Delaroche. We establish an isomorphism concerning the Kasparov equivariant bifunctor RKKG(X;−,−). This isomorphism in K-theory, allows one to extend the half-exactness from the case of the proper algebras (which is analogue to the one obtained by Skandalis in the non-equivariant case) to the case of amenable actions. In particular, we will place ourselves in a significant case, that of hyperbolic displacements of the Poincaré-Lobatschevsky geometry on the unit disc. To cite this article: D. El Morsli, C. R. Acad. Sci. Paris, Ser. I 341 (2005).