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

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- by Søren Eilers, Takeshi Katsura, Mark Tomforde and James West PDF
- Trans. Amer. Math. Soc.
**368**(2016), 3811-3847 Request permission

## 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
- MR Author ID: 609837
- 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
- MR Author ID: 687274
- Email: tomforde@math.uh.edu
**James West**- Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
- Email: jdwest@math.uh.edu
- 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).
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 3811-3847 - MSC (2010): Primary 46L55
- DOI: https://doi.org/10.1090/tran/6443
- MathSciNet review: 3453358