Approximate ideal structures and K-theory.

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 Ktheory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the K¨unneth 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 Künneth formula, then A itself satisfies the Künneth formula. [ABSTRACT FROM AUTHOR]