Geometric Classification of Graph C*-algebras over Finite Graphs

Abstract

Abstract. We address the classi cation problem for graph C * -algebras of nite graphs ( nitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph C * -algebras may come with uncountably many ideals.We nd that in this generality, stable isomorphism of graph C * -algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C * -algebras having isomorphic K-theories. is proves in turn that under this condition, the graph C * -algebras are in fact classi able by K-theory, providing in particular complete classi cation when the C * -algebras in question are either of real rank zero or type I/postliminal. e key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.Our results are applied to discuss the classi cation problem for the quantum lens spaces de ned by Hong and Szymański, and to complete the classi cation of graph C * -algebras associated to all simple graphs with four vertices or less.

Invariance of the Cuntz splice

“…In a similar way as in the above mentioned cases, this result is a key step in the recent development in the geometric classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [ERRS16a,ERRS16b]) as well as in the question of strong classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [CRR17,ERRS16b]). In fact, using the results as well as the proof methods of the present paper, these papers close the classification problem for all Cuntz-Krieger algebras and for all unital graph C * -algebras.…”

Filtered K-Theory for Graph Algebras

“…The authors have shown in [ERRS16] that when classifying graph C * -algebras that do not have real rank 0, it can be useful to replace the full filtered K-theory with a version that only looks at gauge invariant ideals. Motivated by this, we develop a version of ideal related algebraic K-theory relative to a sublattice of ideals.…”

Strong classification of purely infinite Cuntz-Krieger algebras

“…This is used for instance in the proofs of [ERRS16b] to get a classification result. As we are proving a strong classification result, we will need to know more about what this stable isomorphism induces on the reduced filtered K-theory.…”

“…A vertex is a transition state if and only if it is regular and is not the base of a cycle. For a subset S of E 0 , we denote by H(S) the hereditary subset generated by S, and by S the saturation of S (see [ERRS16b,Definition 3.1]).…”

“…Definitions and notation. We adopt verbatim the definitions and the notation from [ERRS16b] -in particular Sections 2, 3 and 4. Since all notation needed is defined and explained in [ERRS16b], we will only recall some of it here, but refer the reader to that paper for the rest.…”