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The ranges of $ K$-theoretic invariants for nonsimple graph algebras


Authors: Søren Eilers, Takeshi Katsura, Mark Tomforde and James West
Journal: Trans. Amer. Math. Soc. 368 (2016), 3811-3847
MSC (2010): Primary 46L55
DOI: https://doi.org/10.1090/tran/6443
Published electronically: August 20, 2015
MathSciNet review: 3453358
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Abstract: There are many classes of nonsimple graph $ C^*$-algebras that are classified by the six-term exact sequence in $ K$-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph $ C^*$-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of $ K$-groups by splicing together smaller graphs whose $ C^*$-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph $ C^*$-algebras.

We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the $ C^*$-algebras under investigation have more than one ideal and where there are currently no relevant classification theories available.


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Additional Information

Søren Eilers
Affiliation: Department for Mathematical Sciences, University of Copenhagen, Universitets- parken 5, DK-2100 Copenhagen Ø, Denmark
Email: eilers@math.ku.dk

Takeshi Katsura
Affiliation: Department of Mathematics, Keio University, Yokohama, 223-8522, Japan
Email: katsura@math.keio.ac.jp

Mark Tomforde
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: tomforde@math.uh.edu

James West
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: jdwest@math.uh.edu

DOI: https://doi.org/10.1090/tran/6443
Keywords: $C^*$-algebras, $K$-theory, six-term exact sequence, classification, range of invariant
Received by editor(s): July 21, 2013
Received by editor(s) in revised form: March 7, 2014
Published electronically: August 20, 2015
Additional Notes: This research was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. Support was also provided by the NordForsk Research Network “Operator Algebras and Dynamics” (grant #11580). The third author was also supported by a grant from the Simons Foundation (#210035).
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