Dynamical complexity and controlled operator K-theory/ E. Guentner, R. Willett & G. Yu

Includes bibliography

Introduction -- Assembly maps -- Groupoids and decompositions -- Controlled K-theory -- Strategy of proof of Theorem 2.11 -- Homotopy invariance -- Mayer-Vietoris -- Finite dynamical complexity for etale groupoids -- Comparison to the Baum-Connes assembly map

In this volume, the authors introduce a property of topological dynamical systems that they call finite dynamical complexity. For systems with this property, one can in principle compute the K-theory of the associated crossed product C^∗-algebra by splitting it up into simpler pieces and using the methods of controlled K-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. The authors have tried to keep the volume as self-contained as possible and hope the main part will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, they 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.