The Ranges of K-theoretic Invariants for Nonsimple Graph Algebras

TL;DR

This paper investigates which cyclic six-term exact sequences in K-theory can be realized by nonsimple graph C*-algebras, introducing a method to construct graphs with prescribed K-theoretic invariants.

Contribution

It develops a general method to construct graphs with specific K-theoretic invariants by splicing smaller graphs, advancing the understanding of the range of K-theoretic invariants for graph C*-algebras.

Findings

Established a method to realize prescribed six-term exact sequences in K-theory.

Obtained the first permanence results for extensions of graph C*-algebras.

Provided tools potentially applicable to more complex C*-algebra classifications.

Abstract

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available.