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 is finite for many interesting examples. In this paper, we use finiteness of dynamic asymptotic dimension of an action to get information on the K-theory of the associated crossed product C*-algebra: specifically, we give a new proof of the Baum-Connes conjecture for such actions. The key tool is controlled K-theory, as developed by Oyono-Oyono and the third author. Our main result is not new: it follows from work of Tu on amenable groupoids. The proof, however, is very different: it amounts to a computation of the K-theory of a crossed product which is quite independent of the topological formula posited by the Baum-Connes machinery. We have tried to keep the paper as self-contained as possible: we hope the main part of the paper will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, we do not assume prior knowledge of controlled K-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant K-theory to set up.