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We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the ‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture.

We also construct explicit maps from the groupoid homology groups to the K‐theory groups of their ‐algebras in degrees zero and one, and investigate their properties.

 

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. 

We introduce a notion of approximate ideal structure for a C*-algebra, and use it as a tool to study K-theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of Oyono-Oyono and Yu in the setting of controlled K-theory of filtered C*-algebras; we do not, however, use that language in this paper. We give two main applications. The first is a vanishing result for K-theory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the Kunneth formula in C*-algebra K-theory: roughly, this says that if A can be decomposed into a pair of subalgebras (C,D) such that C, D, and C∩ D all satisfy the Kunneth formula, then A itself satisfies the Kunneth formula.