On the classification of nonsimple graph C*-algebras

TL;DR

This paper establishes classification results for certain graph C*-algebras with specific ideal structures using K-theory invariants, advancing understanding of their stable isomorphism classes.

Contribution

It introduces a classification method for graph C*-algebras with one or a largest AF-type ideal, based on a general approach involving stability of hereditary subalgebras.

Findings

Classifies graph C*-algebras with one proper nontrivial ideal via K-theory.

Shows that a largest proper ideal being AF leads to similar classification.

Proves that such graph C*-algebras are generally stable except in trivial cases.

Abstract

We prove that a graph C*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C*-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C*-algebra is stable.