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.

This work was started during a sabbatical visit to the University of Munster. I would like to thank the members of the mathematics department there for their warm hospitality.

I would like to particularly thank Clément Dell'Aiera, Dominik Enders, Sabrina Gemsa, Hervé Oyono-Oyono, Ian Putnam, Aaron Tikuisis, Stuart White, Wilhelm Winter, and Guoliang Yu for numerous enlightening conversations relevant to the topics of this paper.

The support of the US NSF through grants DMS 1564281 and DMS 1901522 is gratefully acknowledged.

Finally, my thanks to the anonymous referee for a careful reading of the paper.