Geometric classification of graph $C^{\ast }$-algebras over finite graphs

Abstract

We address the classification problem for graph $C^{\ast }$-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 $C^{\ast }$-algebras may come with uncountable ideal structures. We find that in this generality, stable isomorphism of graph $C^{\ast }$-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 $C^{\ast }$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^{\ast }$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^{\ast }$-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 Szymański, and to complete the classification of graph $C^{\ast }$-algebras associated to all simple graphs with four vertices or less.

Corrected typos, corrected minor errors in statements and proofs of some results, and added Lemma 6.6

References in corpus (5)

• The complete classification of unital graph $C^{\ast }$-algebras: Geometric and strong
• Strong classification of extensions of classifiable C*-algebras
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Cited by in corpus (12)

• The complete classification of unital graph $C^{\ast }$-algebras: Geometric and strong
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• Filtered K-theory for graph algebras
• Strong classification of purely infinite Cuntz-Krieger algebras
• A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras
• Classification conjectures for Leavitt path algebras
• Combinatorial classification of quantum lens spaces
• Geometric classification of isomorphism of unital graph C*-algebras
• Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras
• The spectra of digraphs with Morita equivalent $C^{\ast }$-algebras
• On The Classification of Quantum Lens Spaces of Dimension at most 7