The coarse Baum–Connes conjecture and groupoids. II

Deformation Quantization and the Baum–Connes Conjecture

Alternative titles of this paper would have been “Index theory without index” or “The Baum–Connes conjecture without Baum.” In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields ofC∗-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. Th...

K-théorie Bivariante Pour Les Algèbres De Banach, Groupoïdes Et Conjecture De Baum–connes. Avec Un Appendice D’hervé Oyono-oyono

We construct a KK-theory for Banach algebras, equivariant with respect to the action of a groupoid. We prove the Baum–Connes conjecture with commutative coefficients for hyperbolic groups and for the Poincaré groupoids of foliations with a compact base and a longitudinal Riemannian metrics with negative sectional curvature. Mots clés : théorie de Kasparov ; conjecture de Baum–Connes ; algèbres ...

The Coarse Baum – Connes Conjecture for Spaces Which Admit a Uniform Embedding into Hilbert Space

Higher index theory for certain expanders and Gromov monster groups II

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space X. As discussed in the first paper in this series, this has applications to the Baum-Connes conjecture for ‘Gromov monster’...

Dynamic Asymptotic Dimension and Controlled Operator K - Theory

In earlier work the authors introduced dynamic asymptotic dimension, a notion of dimension for topological dynamical systems that applies to many interesting examples. In this paper, we use finiteness of dynamic asymptotic dimension to get information on the K-theory of the associated crossed product C ̊algebras: specifically, we give a new proof of the Baum-Connes conjecture for such actions. T...