Geometric classification of graph $C^*$-algebras over finite graphs

TL;DR

This paper develops a geometric classification framework for finite graph $C^*$-algebras, extending previous results by relaxing standard conditions and linking stable isomorphism to $K$-theory under certain graph conditions.

Contribution

It introduces a new geometric approach to classify finite graph $C^*$-algebras without the standard (K) condition, relating stable isomorphism to $K$-theory and adjacency matrices.

Findings

Stable isomorphism does not always coincide with Cuntz move equivalence.

Adding a modest condition makes stable isomorphism equivalent to $K$-theory isomorphism.

Complete classification achieved for certain classes of graph $C^*$-algebras.

Abstract

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 (K), so that the graph $C^*$-algebras may come with uncountable ideal structures. 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 quantum lens spaces defined by Hong and Szyma\'nski, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.