Filtered K-Theory for Graph Algebras

  • Søren Eilers
  • Gunnar Restorff
  • Efren Ruiz
  • Adam P. W. Sørensen
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 1)

Abstract

We introduce filtered algebraic K-theory of a ring R relative to a sublattice of ideals. This is done in such a way that filtered algebraic K-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge invariant filtered K-theory for graph algebras. We apply this to verify the Abrams-Tomforde conjecture for a large class of finite graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), by the VILLUM FONDEN through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology, by a grant from the Simons Foundation (# 279369 to Efren Ruiz), and by the Danish Council for Independent Research—Natural Sciences.

References

  1. 1.
    Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005). http://dx.doi.org/10.1016/j.jalgebra.2005.07.028 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abrams, G., Tomforde, M.: Isomorphism and Morita equivalence of graph algebras. Trans. Amer. Math. Soc. 363(7), 3733–3767 (2011). http://dx.doi.org/10.1090/S0002-9947-2011-05264-5 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ara, P., Goodearl, K.R., O’Meara, K.C., Pardo, E.: Separative cancellation for projective modules over exchange rings. Israel J. Math. 105, 105–137 (1998). http://dx.doi.org/10.1007/BF02780325 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ara, P., Moreno, M.A., Pardo, E.: Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10(2), 157–178 (2007). http://dx.doi.org/10.1007/s10468-006-9044-z MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aranda Pino, G., Pardo, E., Siles Molina, M.: Exchange Leavitt path algebras and stable rank. J. Algebra 305(2), 912–936 (2006). http://dx.doi.org/10.1016/j.jalgebra.2005.12.009 MathSciNetCrossRefGoogle Scholar
  6. 6.
    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). http://projecteuclid.org/euclid.ijm/1258138472 MathSciNetMATHGoogle Scholar
  7. 7.
    Cortiñas, G.: Algebraic vs. topological K-theory: a friendly match. In: Topics in Algebraic and Topological K-Theory. Lecture Notes in Mathematics, vol. 2008, pp. 103–165. Springer, Berlin (2011). http://dx.doi.org/10.1007/978-3-642-15708-0_3 MATHGoogle Scholar
  8. 8.
    Drinen, D., Tomforde, M.: The C -algebras of arbitrary graphs. Rocky Mountain J. Math. 35(1), 105–135 (2005).  http://dx.doi.org/10.1216/rmjm/1181069770 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eilers, S., Restorff, G., Ruiz, E., Sørensen, A.P.W.: The complete classification of unital graph C -algebras: geometric and strong. arXiv e-prints (2016, submitted)Google Scholar
  10. 10.
    Eilers, S., Restorff, G., Ruiz, E., Sørensen, A.P.W.: Geometric classification of graph C -algebras over finite graphs. arXiv e-prints Canad. J. Math. (2017). https://cms.math.ca/10.4153/CJM-2017-016-7
  11. 11.
    Fowler, N.J., Laca, M., Raeburn, I.: The C -algebras of infinite graphs. Proc. Amer. Math. Soc. 128(8), 2319–2327 (2000). http://dx.doi.org/10.1090/S0002-9939-99-05378-2 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hay, D., Loving, M., Montgomery, M., Ruiz, E., Todd, K.: Non-stable K-theory for Leavitt path algebras. Rocky Mountain J. Math. 44(6), 1817–1850 (2014).  http://dx.doi.org/10.1216/RMJ-2014-44-6-1817 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jeong, J.A., Park, G.H.: Graph C -algebras with real rank zero. J. Funct. Anal. 188(1), 216–226 (2002).  http://dx.doi.org/10.1006/jfan.2001.3830 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Năstăsescu, C., van Oystaeyen, F.: Graded Ring Theory. North-Holland Mathematical Library, vol. 28. North-Holland Publishing Co., Amsterdam/New York (1982)Google Scholar
  15. 15.
    Raeburn, I.: Graph Algebras. CBMS Regional Conference Series in Mathematics, vol. 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2005)Google Scholar
  16. 16.
    Rangaswamy, K.M.: The theory of prime ideals of Leavitt path algebras over arbitrary graphs. J. Algebra 375, 73–96 (2013). http://dx.doi.org/10.1016/j.jalgebra.2012.11.004 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ruiz, E., Tomforde, M.: Ideal-related K-theory for Leavitt path algebras and graph C -algebras. Indiana Univ. Math. J. 62(5), 1587–1620 (2013).  http://dx.doi.org/10.1512/iumj.2013.62.5123 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ruiz, E., Tomforde, M.: Ideals in graph algebras. Algebr. Represent. Theory 17(3), 849–861 (2014). http://dx.doi.org/10.1007/s10468-013-9421-3 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tomforde, M.: Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra 318(1), 270–299 (2007). http://dx.doi.org/10.1016/j.jalgebra.2007.01.031 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Weibel, C.A.: The K-Book. Graduate Studies in Mathematics, vol. 145. American Mathematical Society, Providence, RI (2013). An introduction to algebraic K-theoryGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Søren Eilers
    • 1
  • Gunnar Restorff
    • 2
  • Efren Ruiz
    • 3
  • Adam P. W. Sørensen
    • 4
  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of Science and TechnologyUniversity of the Faroe IslandsTórshavnFaroe Islands
  3. 3.Department of MathematicsUniversity of Hawaii, HiloHiloUSA
  4. 4.Department of MathematicsUniversity of OsloOsloNorway

Personalised recommendations