K-theory of uniform Roe algebras
We construct a uniform version of the analytic K-homology theory and prove its basic properties such as a Mayer-Vietoris sequence. We show that uniform K-homology is isomorphic to a direct limit of K-theories of certain C*-algebras. Furthermore, we construct an index map (or uniform assembly map) from uniform K-homology into the K-theory of uniform Roe C*-algebras. In an analogy to the coarse Baum--Connes conjecture, this can be viewed as an attempt to provide an algorithm for computing K-theory of uniform Roe algebras. Furthermore, as an application of uniform K-homology, we prove a criterion for amenability. In contrast, we show that uniform Roe C*-algebras of a large class of expanders are not even K-exact. Consequently, their K-theory is in principle not computable by means of exact sequences.