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Geometric Classification of Graph C*-algebras over Finite Graphs

Published online by Cambridge University Press:  20 November 2018

Søren Eilers ,
Gunnar Restorff ,
Efren Ruiz  and
Adam P.W. Sørensen
Søren Eilers
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark e-mail: eilers@math.ku.dk
Gunnar Restorff
Affiliation:
Department of Science and Technology, University of the Faroe Islands, Nóatún h, FO-00 Tórshavn, the Faroe Islands e-mail: gunnarr@setur.fo
Efren Ruiz
Affiliation:
Department of Mathematics, University of Hawaii, Hilo, 200 W.Kawili St., Hilo, Hawaii, 96720-4091 USA e-mail: ruize@hawaii.edu
Adam P.W. Sørensen
Affiliation:
Department of Mathematics, University of Oslo, PO BOX 1053 Blindern, N-0316 Oslo, Norway e-mail: apws@math.uio.no
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Abstract

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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.

Keywords

46L8046L5537B10graph C*-algebrasgeometric classificationK-theoryflow equivalence
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 2 , 01 April 2018 , pp. 294 - 353
Copyright
Copyright © Canadian Mathematical Society 2018

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