Skip to main content
SpringerLink
Log in

Classification of Graph C ∗-Algebras with No More than Four Primitive Ideals

  • Conference paper
  • First Online:
Operator Algebra and Dynamics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 58))

Abstract

We describe the status quo of the classification problem of graph C ∗-algebras with four primitive ideals or less.

Mathematics Subject Classification (2010): 46L80, 19K35.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See Remark 1 for a discussion about the direction of the arrows.

  2. 2.

    The space 4.E was forgotten on page 230 of [29]

  3. 3.

    Although this is not exactly the same definition as the filtrated K-theory in [30], it is known to be the same for all the cases where we have a UCT. For more on this invariant and C ∗-algebrasover X the reader is referred to [30] and the references therein.

  4. 4.

    Here we specify how we view the algebras as algebras over a ← b → c by providing a continuous map from the primitive ideal space to {a, b, c}

  5. 5.

    Here we specify how we view the algebras as algebras over a → b ← c by providing a continuous map from the primitive ideal space to {a, b, c}

References

  1. Arklint, S., Restorff, G., Ruiz, E.: Filtrated K-theory for real rank zero C ∗-algebras. Internat. J. Math. 23(8), 1250,078, 19 (2012). DOI 10.1142/S0129167X12500784. URL http://dx.doi.org/10.1142/S0129167X12500784

  2. Arklint, S.E., Bentmann, R., Katsura, T.: Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras (2013), arXiv:1301.7223v1

    Google Scholar 

  3. Bates, T., Hong, J.H., Raeburn, I., Szymański, W.: The ideal structure of the C ∗-algebras of infinite graphs. Illinois J. Math. 46(4), 1159–1176 (2002). URL http://projecteuclid.org/getRecord?id=euclid.ijm/1258138472

    Google Scholar 

  4. Bentmann, R., Köhler, M.: Universal coefficient theorems for C ∗-algebras over finite topological spaces (2011), arXiv:1101.5702

    Google Scholar 

  5. Blackadar, B.: Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122. Springer-Verlag, Berlin (2006). Theory of C ∗-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III

    Google Scholar 

  6. Brinkmann, G., McKay, B.D.: Posets on up to 16 points. Order 19(2), 147–179 (2002). DOI 10.1023/A:1016543307592. URL http://dx.doi.org/10.1023/A:1016543307592

    Google Scholar 

  7. Brown, L.G.: Stable isomorphism of hereditary subalgebras of C ∗-algebras. Pacific J. Math. 71(2), 335–348 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlsen, T.M., Eilers, S., Tomforde, M.: Index maps in the K-theory of graph algebras. J. K-Theory 9(2), 385–406 (2012). DOI 10.1017/is011004017jkt156. URL http://dx.doi.org/10.1017/is011004017jkt156

  9. Drinen, D., Tomforde, M.: Computing K-theory and Ext for graph C ∗-algebras. Illinois J. Math. 46(1), 81–91 (2002). URL http://projecteuclid.org/getRecord?id=euclid.ijm/1258136141

  10. Drinen, D., Tomforde, M.: The C ∗-algebras of arbitrary graphs. Rocky Mountain J. Math. 35(1), 105–135 (2005). DOI 10.1216/rmjm/1181069770. URL http://dx.doi.org/10.1216/rmjm/1181069770

    Google Scholar 

  11. Eilers, S., Katsura, T.: Semiprojectivity and properly infinite projections in graph C ∗-algebra. In preparation

    Google Scholar 

  12. Eilers, S., Loring, T.A., Pedersen, G.K.: Morphisms of extensions of C ∗-algebras: pushing forward the Busby invariant. Adv. Math. 147(1), 74–109 (1999). DOI 10.1006/aima.1999.1834. URL http://dx.doi.org/10.1006/aima.1999.1834

    Google Scholar 

  13. Eilers, S., Restorff, G., Ruiz, E.: Classification of extensions of classifiable C ∗-algebras. Adv. Math. 222(6), 2153–2172 (2009). DOI 10.1016/j.aim.2009.07.014. URL http://dx.doi.org/10.1016/j.aim.2009.07.014

    Google Scholar 

  14. Eilers, S., Restorff, G., Ruiz, E.: On graph C ∗-algebras with a linear ideal lattice. Bull. Malays. Math. Sci. Soc. (2) 33(2), 233–241 (2010)

    Google Scholar 

  15. Eilers, S., Restorff, G., Ruiz, E.: Classifying C ∗-algebras with both finite and infinite subquotients. J. Funct. Anal. (2013). DOI 10.1016/j.jfa.2013.05.006. In press

    Google Scholar 

  16. Eilers, S., Restorff, G., Ruiz, E.: The ordered K-theory of a full extension. Canad. J. Math. (2013). DOI 10.4153/CJM-2013-015-7. In press

    Google Scholar 

  17. Eilers, S., Restorff, G., Ruiz, E.: Strong classification of extensions of classifiable C ∗-algebras (2013), arXiv:1301.7695v1

    Google Scholar 

  18. Eilers, S., Ruiz, E., Sørensen, A.P.W.: Amplified graph C ∗-algebras. Münster J. Math. 5, 121–150 (2012)

    Google Scholar 

  19. Eilers, S., Tomforde, M.: On the classification of nonsimple graph C ∗-algebras. Math. Ann. 346(2), 393–418 (2010). DOI 10.1007/s00208-009-0403-z. URL http://dx.doi.org/10.1007/s00208-009-0403-z

  20. Elliott, G.A.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38(1), 29–44 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elliott, G.A.: Towards a theory of classification. Adv. Math. 223(1), 30–48 (2010). DOI 10.1016/j.aim.2009.07.018. URL http://dx.doi.org/10.1016/j.aim.2009.07.018

  22. Elliott, G.A., Kucerovsky, D.: An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem. Pacific J. Math. 198(2), 385–409 (2001). DOI 10.2140/pjm.2001.198.385. URL http://dx.doi.org/10.2140/pjm.2001.198.385

    Google Scholar 

  23. Jeong, J.A.: Real rank of C ∗-algebras associated with graphs. J. Aust. Math. Soc. 77(1), 141–147 (2004). DOI 10.1017/S1446788700010211. URL http://dx.doi.org/10.1017/S1446788700010211

    Google Scholar 

  24. Kirchberg, E.: The classification of purely infinite C ∗-algebras using Kasparov’s theory (1994). Preprint, third draft

    Google Scholar 

  25. Kirchberg, E.: Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren. In: C ∗-algebras (Münster, 1999), pp. 92–141. Springer, Berlin (2000)

    Google Scholar 

  26. Kirchberg, E., Rørdam, M.: Non-simple purely infinite C ∗-algebras. Amer. J. Math. 122(3), 637–666 (2000). URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.3kirchberg.pdf

    Google Scholar 

  27. Kirchberg, E., Rørdam, M.: Infinite non-simple C ∗-algebras: absorbing the Cuntz algebras \(\mathcal{O}_{\infty }\). Adv. Math. 167(2), 195–264 (2002). DOI 10.1006/aima.2001.2041. URL http://dx.doi.org/10.1006/aima.2001.2041

    Google Scholar 

  28. Kucerovsky, D., Ng, P.W.: The corona factorization property and approximate unitary equivalence. Houston J. Math. 32(2), 531–550 (electronic) (2006)

    Google Scholar 

  29. Meyer, R., Nest, R.: C ∗-algebras over topological spaces: the bootstrap class. Münster J. Math. 2, 215–252 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Meyer, R., Nest, R.: C∗-algebras over topological spaces: filtrated K-theory. Canad. J. Math. 64(2), 368–408 (2012). DOI 10.4153/CJM-2011-061-x. URL http://dx.doi.org/10.4153/CJM-2011-061-x

    Google Scholar 

  31. Murphy, G.J.: C ∗-algebras and operator theory. Academic Press Inc., Boston, MA (1990)

    MATH  Google Scholar 

  32. Pedersen, G.K.: Pullback and pushout constructions in C ∗-algebra theory. J. Funct. Anal. 167(2), 243–344 (1999). DOI 10.1006/jfan.1999.3456. URL http://dx.doi.org/10.1006/jfan.1999.3456

    Google Scholar 

  33. Phillips, N.C.: A classification theorem for nuclear purely infinite simple C ∗-algebras. Doc. Math. 5, 49–114 (electronic) (2000)

    Google Scholar 

  34. Raeburn, I.: Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2005)

    Google Scholar 

  35. Restorff, G.: Classification of non-simple C ∗-algebras. Ph.D. thesis, University of Copenhagen (2008). URL http://www.math.ku.dk/~restorff/papers/thesis.pdf

  36. Rørdam, M.: Classification of extensions of certain C ∗-algebras by their six term exact sequences in K-theory. Math. Ann. 308(1), 93–117 (1997). DOI 10.1007/s002080050067. URL http://dx.doi.org/10.1007/s002080050067

  37. Tomforde, M.: Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra 318(1), 270–299 (2007). DOI 10.1016/j.jalgebra.2007.01.031. URL http://dx.doi.org/10.1016/j.jalgebra.2007.01.031

    Google Scholar 

  38. Toms, A.S., Winter, W.: Strongly self-absorbing C ∗-algebras. Trans. Amer. Math. Soc. 359(8), 3999–4029 (2007). DOI 10.1090/S0002-9947-07-04173-6. URL http://dx.doi.org/10.1090/S0002-9947-07-04173-6

    Google Scholar 

  39. Zhang, S.: K 1-groups, quasidiagonality, and interpolation by multiplier projections. Trans. Amer. Math. Soc. 325(2), 793–818 (1991). DOI 10.2307/2001648. URL http://dx.doi.org/10.2307/2001648

    Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge support from the NordForsk Research Network “Operator Algebra and Dynamics” (grant #11580) and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen, Denmark

    Søren Eilers

  2. Department of Science and Technology, University of the Faroe Islands, Nóatún 3, FO-100, Tórshavn, Faroe Islands

    Gunnar Restorff

  3. Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, HI, 96720-4091, USA

    Efren Ruiz

Authors
  1. Søren Eilers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gunnar Restorff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Efren Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Søren Eilers .

Editor information

Editors and Affiliations

  1. Dept. of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Toke M. Carlsen

  2. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Søren Eilers

  3. Dept. of Science and Technology, University of the Faroe Islands, Tórshavn, Faroe Islands

    Gunnar Restorff

  4. Culture and Communication (UKK) Division of Applied Mathematics The School of Education, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Eilers, S., Restorff, G., Ruiz, E. (2013). Classification of Graph C ∗-Algebras with No More than Four Primitive Ideals. In: Carlsen, T., Eilers, S., Restorff, G., Silvestrov, S. (eds) Operator Algebra and Dynamics. Springer Proceedings in Mathematics & Statistics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39459-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation