CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Geometric classification of graph C*-algebras over finite graphs

  • 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
Format:   LaTeX   MathJax   PDF  

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 graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
MSC Classifications: 46L35, 46L80, 46L55, 37B10 show english descriptions Classifications of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Symbolic dynamics [See also 37Cxx, 37Dxx]
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]
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/