Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-18T08:36:08.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras

Published online by Cambridge University Press:  20 October 2014

Sara E. Arklint ,
Rasmus Bentmann  and
Takeshi Katsura
Sara E. Arklint
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, arklint@math.ku.dk
Rasmus Bentmann
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, bentmann@math.ku.dk
Takeshi Katsura
Affiliation:
Department of Mathematics, Keio university, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan, katsura@math.keio.ac.jp
Get access

Abstract

We show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we give a characterization of purely infinite Cuntz–Krieger algebras whose primitive ideal space is an accordion space.

Keywords

C*-algebrasgraph C*-algebrasclassificationfiltered K-theoryreal rank zero46L3546L80(46L55)
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 570 - 613
Copyright
Copyright © ISOPP 2014 

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.Abrams, Gene, Tomforde, Mark, Isomorphism and Morita equivalence of graph algebras Trans. Amer. Math. Soc. 363 (2011), 3733–3767.CrossRefGoogle Scholar
2.Arklint, Sara E., Do phantom Cuntz-Krieger algebras exist? Operator Algebra and Dynamics (Faroe Islands, 2012) 31–40, Springer Proceedings in Mathematics & Statistics 58, Springer, Berlin.Google Scholar
3.Arklint, Sara E., Bentmann, Rasmus, Katsura, Takeshi, The K-theoretical range of Cuntz-Krieger algebras, J. Funct. Anal. 266(8) (2014), 5448–5466.CrossRefGoogle Scholar
4.Arklint, Sara E., Restorff, Gunnar, Ruiz, Efren, Filtrated K-theory of real rank zero C*-algebras, Internat. J. Math. 23(8) (2012), 1250078, 19.Google Scholar
5.Arklint, Sara E., Ruiz, Efren, Corners of Cuntz–Krieger algebras, arXiv:1209.4336, 2012.Google Scholar
6.Bentmann, Rasmus, Filtrated K-theory and classification of C*-algebras, Diplom thesis, University of Göttingen, 2010, http://www.uni-math.gwdg.de/rbentma/diplom_thesis.pdf.Google Scholar
7.Bentmann, Rasmus, Kirchberg X-algebras with real rank zero and intermediate cancellation, arXiv:math/1301.6652, 2013, to appear in J. Noncommut. Geom.Google Scholar
8.Bentmann, Rasmus, Köhler, Manuel, Universal Coefficient Theorems for C*-algebras over finite topological spaces, arXiv:math/1101.5702v3, 2011.Google Scholar
9.Boyle, Mike, Huang, Danrun, Poset block equivalence of integral matrices, Trans. Amer. Math. Soc. 355(10) (2003), 3861–3886 (electronic).CrossRefGoogle Scholar
10.Brown, Lawrence G., Pedersen, Gert K., C*-algebras of real rank zero, J. Funct. Anal. 99(1) (1991), 131–149.Google Scholar
11.Cuntz, Joachim, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57(2) (1977), 173–185.Google Scholar
12.Cuntz, Joachim, Krieger, Wolfgang, Aclass of C*-algebras and topological Markov chains, Invent. Math. 56(3) (1980), 251–268.Google Scholar
13.Søren, Eilers, Restorff, Gunnar, Ruiz, Efren, Classifying C*-algebras with both finite and infinite subquotients, J. Funct. Anal. 265(3) (2013), 449–468.Google Scholar
14.Søren, Eilers, Restorff, Gunnar, Ruiz, Efren, Strong classification of extensions of classifiable C*-algebras, arXiv:1301.7695v1, 2013.Google Scholar
15.Hong, Jeong Hee, WojciechSzymański, Purely infinite Cuntz-Krieger algebras of directed graphs, Bull. London Math. Soc. 35(5) (2003), 689–696.Google Scholar
16.Kirchberg, Eberhard, The classification of Purely Infinite C*-algebras using Kasparov's Theory, to appear in the Fields Institute Communication series.Google Scholar
17.Kirchberg, Eberhard, Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren, C*-algebras (Münster, 1999), pp. 92–141, Springer, Berlin, 2000.Google Scholar
18.Lin, Hua Xin, Rørdam, Mikael, Extensions of inductive limits of circle algebras, J. London Math. Soc. (2) 51(3) (1995), 603–613.CrossRefGoogle Scholar
19.Meyer, Ralf, Nest, Ryszard, C*-algebras over topological spaces: filtrated K-theory, Canad. J. Math. 64(2) (2012), 368–408.Google Scholar
20.Meyer, Ralf, Nest, Ryszard, C *-algebras over topological spaces: the bootstrap class, Münster J. Math. 2 (2009), 215–252.Google Scholar
21.Pasnicu, Cornel, Rørdam, Mikael, Purely infinite C *-algebras of real rank zero, J. Reine Angew. Math. 613 (2007), 51–73.Google Scholar
22.Phillips, N. Christopher, A classification theorem for nuclear purely infinite simple C*-algebras, Doc. Math. 5 (2000), 49–114 (electronic).CrossRefGoogle Scholar
23.Restorff, Gunnar, Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math. 598 (2006), 185–210.Google Scholar
24.Restorff, Gunnar, Ruiz, Efren, On Rørdam's classification of certain C*-algebras with one non-trivial ideal. II, Math. Scand. 101(2) (2007), 280–292.Google Scholar
25.Rørdam, Mikael, Classification of Cuntz-Krieger algebras, K-Theory 9(1) (1995), 31–58.Google Scholar
26.Rørdam, Mikael, Classification of extensions of certain C*-algebras by their six term exact sequences in K-theory, Math. Ann. 308(1) (1997), 93–117.Google Scholar
27. The Stacks Project Authors, Stacks Project, available online at: http://math.columbia.edu/algebraic_geometry/stacks-gi.Google Scholar
28.Toms, Andrew S., Winter, Wilhelm, Strongly self-absorbing C*-algebras, Trans. Amer. Math. Soc. 359(8) (2007), 3999–4029.CrossRefGoogle Scholar