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Geometric classification of graph C*algebras over finite graphs


Published:20170901
Printed: Apr 2018
Søren Eilers,
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK2100 Copenhagen, Denmark
Gunnar Restorff,
Department of Science and Technology, University of the Faroe Islands, Nóatún 3, FO100 Tórshavn, the Faroe Islands
Efren Ruiz,
Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 967204091 USA
Adam P. W. Sørensen,
Department of Mathematics, University of Oslo, PO BOX 1053 Blindern, N0316 Oslo, Norway
Abstract
We address the classification problem for graph $C^*$algebras of
finite graphs (finitely many edges and vertices), containing
the class of CuntzKrieger 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.
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]
