On the classification of nonsimple graph C *-algebras

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.

Index maps in theK-theory of graph algebras

“…having only the trivial automorphism, so the computations combine with [7,Theorem 4.7] to show that C * (F y,z ) ⊗ K ≃ C * (F y ′ ,z ′ ) ⊗ K precisely when 4 − z = ±(4 − z ′ ) and y − y ′ ∈ (4 − z)Z.…”

“…is a complete stable isomorphism invariant for a graph C * -algebra C * (E) of real rank zero containing a proper nontrivial ideal J when any of the following are satisfied • J is the unique proper nontrivial ideal of C * (E) ( [7,Theorem 4.5]),…”

“…• J is a smallest proper nontrivial ideal of C * (E), and C * (E)/J is AF ([6, Corollary 6.4]), • J is a largest proper nontrivial ideal of C * (E), and J is AF ( [7,Theorem 4.7]). …”

Classification of Graph C ∗-Algebras with No More than Four Primitive Ideals

“…The separated case. The classification problem for the two mixed cases with | Prim(A)| = 2 not covered by the results mentioned above -the tempered signatures 2.1.1 and 2.1.2 -were resolved in [ET10] drawing heavily on [ERR09]. In [ERRa], we generalized this to more complicated cases having the separation property which is automatic in the two-point case, as detailed below.…”

The suspension of a graph, and associated C⁎-algebras

“…Each finite directed graph E determines a corresponding 1-sided shift space (X E , σ). Cuntz-Krieger algebras [5], and more generally graph C * -algebras [8,18,19], encode the dynamics (X E , σ) C * -algebraically, and there has been intense interest in these C *algebras ever since their introduction-see, for example [3,4,6,7,10,30] and the bibliography of [25].…”