The coarse Baum-Connes conjecture and controlled operator K-theory

Abstract

The Atiyah-Singer index theorem has been vastly generaized to higher index theory for elliptic operators in the context of noncommutative geometry. Higher index theory has important applications in differential topology and differential geometry such as the Novikov conjecture on homotopy invariance of higher signature and the existence preoblem of Riemannian metrics with positive scalar curvature. The Baum-Connes conjecture and the coarse Baum-Connes conjecture are algorithms to compute the higher indices of elliptic differential operators. The controlled operator K-theory, a refined version of classical operator K-theory, is a powerful tool to study these conjectures. In this paper, we study the controlled operator K-theory, give a characterization of the K-theory elements in the image of the Baum-Connes map, and verify the coarse Baum-Connes conjecture for a large class of spaces, especially for spaces with finite decomposition complexities.