Algebraic -theory of Fredholm modules and -theory.
Bivariant -theory for locally convex algebras and the Chern-Connes character. (Bivariante -Theorie für lokalconvexe Algebren und der Chern-Connes-Charakter.)
Coarse topology, enlargeability, and essentialness
Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the -theory of the corresponding reduced -algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.
Equivariant Euler characteristics and -homology Euler classes for proper cocompact -manifolds.
Equivariant KK-theory for semimultiplicative sets.
Equivariant local cyclic homology and the equivariant Chern-Connes character.
Geometric -homolgy and controlled paths.
Group Cohomology with Lipschitz Control and Higher Signatures.
Homotopy invariance of higher signatures and -manifold groups
For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable -manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable -manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The...
-equivalence and continuous bundles of -algebras.
-theory as the -theory of -categories.
-théorie algébrique négative et -théorie topologique de l’algèbre de Fréchet des opérateurs régularisants
KK-theory of the full free product of unital C*-algebras.
Non-commutative differential geometry
On the K-theory of the -algebra generated by the left regular representation of an Ore semigroup
We compute the -theory of -algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the -theory of these semigroup -algebras in terms of the -theory for the reduced group -algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.
Operator k-theory for the group SU(n, 1).
Progrès récents sur la conjecture de Baum-Connes. Contribution de Vincent Lafforgue
Projections in full C*-algebras of semisimple Lie groups.
Quantum SU(2) and the Baum-Connes conjecture
We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.
Real versus complex K-theory using Kasparov's bivariant KK-theory.