The ranges of 𝐾-theoretic invariants for nonsimple graph algebras

Abstract

There are many classes of nonsimple graph C ∗ C^* -algebras that are classified by the six-term exact sequence in K 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 ∗ 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 K -groups by splicing together smaller graphs whose C ∗ 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 ∗ 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 ∗ C^* -algebras under investigation have more than one ideal and where there are currently no relevant classification theories available.