Hervé Oyono-Oyono

Baum–Connes Conjecture and Group Actions on Trees

Let A be a separable C∗−algebra and Γ be a discrete and countable group acting on A by automorphisms. The Baum-Connes conjecture with coefficients relates the K−theory group of AoΓ, the reduced crossed product of A by Γ to the topology of a space EΓ canonically associated (up to equivariant homotopy) with the group Γ and to the K−theory group of the crossed product of A by the finite subgroups of Γ. The space EΓ is the universal example for proper actions of Γ (see [3], definition 1.6). More precisely, if Z is a metric space with a (metrically) proper action of Γ, we define the equivariant K−homology of Z with coefficients in A by

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups

Fibrations with noncommutative fibers

K^{G}_{*}(X)

Calcul du label des gaps pour les quasi-cristaux

, using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the “official” equivariant K-homology groups) and show that these are isomorphisms.

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C � -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. In recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent role in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C0(X)-algebra in the sense of Kasparov (see (17)): it is a C*-algebra A together with a non-degenerate ∗-homomorphism

Principal non-commutative torus bundles

We give in the present Note a proof of the Bellissard gap-labelling conjecture for quasi-crystals. Our main tools are the measured index theorem for laminations on the one hand, and the naturality of the longitudinal Chern character on the other hand. To cite this article: M.-T. Benameur, H. Oyono-Oyono, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 667–670.