Dynamical complexity and controlled operator K-theory

Abstract

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the $K$-theory of the associated crossed product $C^{\ast }$-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. We have tried to keep the paper as self-contained as possible: we hope the main part 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.

Second version has extensive revisions: chiefly, the main argument has been reworked based on a new notion of `finite dynamical complexity'. This is partly to correct an error, and partly as the new version seemed simpler and more natural. A new appendix has been added comparing finite dynamical complexity to other properties. Third version corrects a minor error in Definition 7.4

References in corpus (1)

• Some notes on property A

Cited by in corpus (6)

• Going-Down functors and the Künneth formula for crossed products by étale groupoids
• The equivariant coarse Novikov conjecture and coarse embedding
• Equivariant Coarse (Co-)Homology Theories
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• Nuclear dimension of subhomogeneous twisted groupoid C*-algebras and dynamic asymptotic dimension
• Expanders are counterexamples to the ${\ell }^p$ coarse Baum-Connes conjecture