Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T10:02:36.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Index maps in the K-theory of graph algebras

Published online by Cambridge University Press:  17 May 2011

Toke Meier Carlsen ,
Søren Eilers  and
Mark Tomforde
Toke Meier Carlsen
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, [email protected]
Søren Eilers
Affiliation:
Department for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, [email protected]
Mark Tomforde
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, [email protected]
Get access

Abstract

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension

in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.

Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

Keywords

Graph C*-algebrasclassificationextensionsK-theory
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 385 - 406
Copyright
Copyright © ISOPP 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ara, P., Moreno, M. A., and Pardo, E.. Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10(2):157–178, 2007.CrossRefGoogle Scholar
2.Bates, T., Hong, J. H., Raeburn, I., and Szymański, W., The ideal structure of the C*-algebras of infinite graphs. Illinois J. Math 46:1159–1176, 2002.CrossRefGoogle Scholar
3.Bates, T., Pask, D., Raeburn, I., and Szymański, W., The C*-algebras of row-finite graphs New York J. Math. 6:307–324, 2000.Google Scholar
4.Brown, L.G. and Pedersen, G.K.. C*-algebras of real rank zero. J. Funct. Anal. 99:131–149, 1991.CrossRefGoogle Scholar
5.Cuntz, J.. On the homotopy groups of the space of endomorphisms of a C*-algebra (with applications to topological Markov chains). In Operator algebras and group representations, Vol. I (Neptun, 1980), Monogr. Stud. Math. 17, pages 124–137. Pitman, Boston, MA, 1984.Google Scholar
6.Drinen, D. and Tomforde, M.. Computing K-theory and Ext for graph C*-algebras. Illinois J. Math. 46:81–91, 2002.CrossRefGoogle Scholar
7.Eilers, S., Restorff, G., and Ruiz, E.. Classifying C*-algebras with both finite and infinite subquotients. Submitted for publication. arXiv:1009.4778v1Google Scholar
8.Eilers, S. and Tomforde, M.. On the classification of nonsimple graph algebras. Math. Ann. 346:393–418, 2010.CrossRefGoogle Scholar
9.Fowler, N.J. and Raeburn, I.. The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48(1):155–181, 1999.CrossRefGoogle Scholar
10.Jeong, J.A. Real rank of C*-algebras associated with graphs. J. Aust. Math. Soc. 77(1):141–147, 2004.CrossRefGoogle Scholar
11.Jeong, J.A and Park, G. H.. Graph C*-algebras with real rank zero. J. Funct. Anal. 188(1):216–226, 2002.CrossRefGoogle Scholar
12.Katsura, T.. On C*-algebras associated with C*-correspondences. J. Funct. Anal. 217(2):366–401, 2004.CrossRefGoogle Scholar
13.Kumjian, A., Pask, D., Raeburn, I., and Renault, J.. Graphs, groupoids, and Cuntz-Krieger algebras. J. Funct. Anal. 144:505–541, 1997.CrossRefGoogle Scholar
14.Muhly, P.S. and Tomforde, M.. Adding tails to C*-correspondences. Doc. Math. 9:79–106, 2004.CrossRefGoogle Scholar
15.Paschke, W.. K-theory for actions of the circle group on C*-algebras. J. Operator Theory 6(1):125–133, 1981.Google Scholar
16.Pask, D. and Raeburn, I.. On the K-theory of Cuntz-Krieger algebras. Publ. RIMS, Kyoto Univ. 32: 415–443, 1996.CrossRefGoogle Scholar
17.Raeburn, I. and Szymański, W.. Cuntz-Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc. 356:39–59, 2004.CrossRefGoogle Scholar
18.Rørdam, M.. Classification of Cuntz-Krieger algebras. K-Theory 9(1):31–58, 1995.CrossRefGoogle Scholar
19.Rørdam, M., Larsen, F., and Laustsen, N.. An introduction to K-theory for C*-algebras, London Mathematical Society Student Texts 49, Cambridge University Press, Cambridge, 2000.Google Scholar
20.Tomforde, M.. The ordered K 0-group of a graph C*-algebra. C. R. Math. Acad. Sci. Soc. R. Can. 25(1):19–25, 2003.Google Scholar