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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*-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. 

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*-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. 

We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras over metric spaces that are coarse unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent. 

A C*-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's KK-theory to a commutative C*-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear C*-algebras. We introduce the idea of a C* -algebra that "decomposes" over a class C of C*-algebras. Roughly, this means that locally there are approximately central elements that approximately cut the C*-algebra into two C*-subalgebras from C that have well-behaved intersection. We show that if a C* -algebra decomposes over the class of nuclear, UCT C*-algebras, then it satisfies the UCT. The argument is based on a Mayer–Vietoris principle in the framework of controlled KK-theory; the latter was introduced by the authors in an earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear C*-algebras are amenable. We say that a C*-algebra has finite complexity if it is in the smallest class of C*-algebras containing the finite-dimensional C*-algebras, and closed under decomposability; our main result implies that all C*-algebras in this class satisfy the UCT. The class of C*-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a C*-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear C*-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear C*-algebras. 

The universal coefficient theorem is a fundamental problem in classification of C*-algebras. We develop quantitative operator K-homology theory to prove the universal coefficient theorem for C*-algebras with finite complexity. 

The universal coefficient theorem is a fundamental problem in the classification theory for C*-algebras. In this article we develop a quantitative K-homology theory to prove the universal coefficient theorem for C*-algebras with finite complexity. 

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated with a uniformly locally finite metric space X. Under weak assumptions, these C*-algebras contain embedded copies of certain matrix algebras. We aim to show they cannot contain any other von Neumann algebras. One of our main results shows that the only embedded von Neumann algebras are the “obvious” ones. 

Complexity rank for C*-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most n if you can repeatedly cut the C∗-algebra in half at most n times, and end up with something finite-dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as “two-colored local finite-dimensionality”. We first show that, for separable, unital, and simple C*-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear C*-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero K-theory groups. However, we also show using a K-theoretic obstruction (torsion in K1) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg–Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it equals one when the K1-group is torsion-free, and equals two otherwise.