Geometric Classification of Graph C-algebras over Finite Graphs

We address the classification problem for graph ${{C}^{*}}$ -algebras of finite graphs (finitely 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 $\left( K \right)$ , so that the graph ${{C}^{*}}$ -algebras may come with uncountably many ideals. We find 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. This proves in turn that under this condition, the graph ${{C}^{*}}$ -algebras are in fact classifiable by $K$ -theory, providing, in particular, complete classification when the ${{C}^{*}}$ - algebras in question are either of real rank zero or type I/postliminal. The 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 classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph ${{C}^{*}}$ -algebras associated with all simple graphs with four vertices or less.