[Paper Review] Approximate ideal structures and K-theory

Rufus Willett|arXiv (Cornell University)|Aug 25, 2019

Advanced Topology and Set Theory28 references4 citations

TL;DR

This paper introduces approximate ideal structures in C*-algebras to study K-theory, generalizing the Mayer-Vietoris sequence and extending tools from nuclear dimension and dynamical complexity. It proves a vanishing theorem for K-theory relevant to the Baum-Connes conjecture and establishes a permanence result for the Künneth formula: if a C*-algebra decomposes into subalgebras satisfying the Künneth formula, so does the whole algebra.

ABSTRACT

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 Künneth 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 $C\cap D$ all satisfy the Künneth formula, then $A$ itself satisfies the Künneth formula.

Motivation & Objective

• To develop a new framework—approximate ideal structures—for analyzing K-theory in C*-algebras.
• To generalize classical tools like the Mayer-Vietoris sequence to non-ideal decompositions.
• To establish conditions under which the Künneth formula holds for C*-algebras via decomposition.
• To provide a K-theory vanishing result relevant to the Baum-Connes conjecture.
• To connect abstract K-theory tools with geometric and dynamical complexity in groupoid C*-algebras.

Proposed method

• Introduces the notion of approximate ideal structures as a generalization of ideal decompositions in C*-algebras.
• Uses the boundary map in K-theory to define a long exact sequence-like structure even when ideals are not strictly present.
• Applies techniques inspired by nuclear dimension and finite dynamical complexity to control K-theory behavior.
• Employs the summation and product maps in K-theory to analyze exactness and injectivity/surjectivity properties.
• Leverages conditional expectations and tensor product functors to construct approximate lifts in the absence of exact ideals.
• Applies the framework to groupoid C*-algebras, particularly étale groupoids, to derive permanence results.

Experimental results

Research questions

• RQ1 Under what conditions can the K-theory of a C*-algebra be controlled via approximate ideal decompositions rather than exact ideals?
• RQ2 How can the Mayer-Vietoris-type long exact sequence be generalized beyond ideal decompositions?
• RQ3 When does the Künneth formula for K-theory persist under algebraic decompositions of a C*-algebra?
• RQ4 What role do dynamical complexity and nuclear dimension play in K-theory vanishing results?
• RQ5 Can the Baum-Connes conjecture be approached via approximate ideal structures and boundary class analysis?

Key findings

• A vanishing theorem for K-theory is established, providing a condition under which K-groups vanish in a way relevant to the Baum-Connes conjecture.
• The paper proves a permanence result: if a C*-algebra A decomposes into subalgebras C, D, and C ∩ D all satisfying the Künneth formula, then A itself satisfies the Künneth formula.
• For ample, second countable, locally compact, Hausdorff étale groupoids, the class of clopen subgroupoids satisfying the Künneth formula is closed under decomposability.
• If a groupoid has strong finite dynamical complexity, then its reduced C*-algebra satisfies the Künneth formula.
• The uniform Roe algebra of a bounded geometry metric space with finite decomposition complexity satisfies the Künneth formula, as it arises from a groupoid with strong finite dynamical complexity.