Expanders, Exact Crossed Products, and the Baum-connes Conjecture
نویسندگان
چکیده
Abstract. We reformulate the Baum-Connes conjecture with coe cients by introducing a new crossed product functor for C⇤-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.
منابع مشابه
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’...
متن کاملThe Baum-connes Conjecture via Localization of Categories
We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this mor...
متن کاملFinite group extensions and the Baum-Connes conjecture
In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups. The Baum-Connes conjecture (in this note the term will always m...
متن کاملDynamic 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...
متن کاملThe Baum-connes Conjecture for Hyperbolic Groups
The Baum-Connes conjecture states that, for a discrete group G, the K-homology groups of the classifying space for proper G-action is isomorphic to the K-groups of the reduced group C-algebra of G [3, 2]. A positive answer to the Baum-Connes conjecture would provide a complete solution to the problem of computing higher indices of elliptic operators on compact manifolds. The rational injectivit...
متن کامل