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# Geometric classification of graph C*-algebras over finite graphs

Published:2017-09-01

• Søren Eilers,
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
• Gunnar Restorff,
Department of Science and Technology, University of the Faroe Islands, Nóatún 3, FO-100 Tórshavn, the Faroe Islands
• Efren Ruiz,
Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 96720-4091 USA
• Adam P. W. Sørensen,
Department of Mathematics, University of Oslo, PO BOX 1053 Blindern, N-0316 Oslo, Norway
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## 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 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 quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.
 Keywords: graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 37B10 - Symbolic dynamics [See also 37Cxx, 37Dxx]

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