On the coarse Baum-Connes conjecture

Abstract

The Baum-Connes conjecture [2, 3] concerns the K-theory of the reduced group C-algebra C r (G) for a locally compact group G. One can define a map from the equivariant K-homology of the universal proper G-space EG to K∗(C ∗ r (G)): each K-homology class defines an index problem, and the map associates to each such problem its analytic index. The conjecture is that this map is an isomorphism. The injectivity of the map has geometric and topological consequences, implying the Novikov conjecture for example; the surjectivity has consequences for C-algebra theory and is related to problems in harmonic analysis. In geometric topology it has proved to be very useful to move from studying classical surgery problems on a compact manifold M to studying bounded surgery problems over its universal cover (see [8, 24] for example). In terms of L-theory, one replaces the classical L-theory of Zπ by the L-theory, bounded over |π|, of Z (here |π| denotes π considered as a metric space, with a word length metric). Now the authors, motivated by considerations of index theory on open manifolds, have studied a C-algebra C(X) associated to any proper metric space X , and it has recently become quite clear that the passage from C r (π) to C (|π|) is an analytic version of the same geometric idea. Moreover, various descent arguments have been given [4, 9, 5, 17, 27], both in the topological and analytic contexts, which show that a ‘sufficiently canonical’ proof of an analogue of the Baum-Connes conjecture in the bounded category will imply the classical version of the Novikov conjecture. The purpose of this paper is to give a precise formulation of the Baum-Connes conjecture for the C-algebras C(X) (filling in the details of the hints in the last section of [26]) and to prove the conjecture for spaces which are non-positively curved in some sense, including affine buildings and hyperbolic metric spaces in the sense of Gromov. Notice that while the classical Novikov conjecture has been established for the analogous class of groups, the Baum-Connes conjecture has not. Unfortunately there does not seem to be any descent principle for the surjectivity side of the Baum-Connes conjecture as there is for the injectivity side. The main tool that we will use in this paper is the invariance of K∗(C (X)) under coarse homotopy, established by the authors in [15]. Coarse homotopy is a rather weak equivalence relation on metric spaces, weak enough that (for example) many spaces are coarse homotopy equivalent to open cones OY on compact spaces Y . The idea of ‘reduction to a cone on an ideal boundary’ is also used in some topological approaches to the Novikov conjecture, but the notion of coarse homotopy