Algebras and Representation Theory

, Volume 10, Issue 2, pp 157–178

Nonstable K-theory for Graph Algebras



We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of LK(E) and the lattice of order-ideals of V(LK(E)). When K is the field \(\mathbb C\) of complex numbers, the algebra \(L_{\mathbb C}(E)\) is a dense subalgebra of the graph C*-algebra C*(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

Key words

graph algebra weak cancellation separative cancellation refinement monoid nonstable K-theory ideal lattice 

Mathematics Subject Classifications (2000)

Primary 16D70 46L35 Secondary 06A12 06F05 46L80 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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)MATHMathSciNetGoogle Scholar
  3. 3.
    Ara, P., Goodearl, K.R., O’Meara, K.C., Raphael, R.: K 1 of separative exchange rings and C*-algebras with real rank zero. Pacific J. Math. 195, 261–275 (2000)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ara, P., Facchini, A.: Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math. 18, 365–389 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bates, T., Pask, D., Raeburn, I., Szymański, W.: The C*-algebras of row-finite graphs. New York J. Math. 6, 307–324 (2000)MATHMathSciNetGoogle Scholar
  6. 6.
    Bergman, G.M.: Coproducts and some universal ring constructions. Trans. Amer. Math. Soc. 200, 33–88 (1974)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blackadar, B.: Rational C*-algebras and nonstable K-theory. Rocky Mountain J. Math. 20(2), 285–316 (1990)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Blackadar, B.: K-theory for operator algebras, 2nd edn. In: Mathematical Science Research Institute Publications, vol. 5. Cambridge University Press, Massachussetts (1998)Google Scholar
  9. 9.
    Brookfield, G.: Cancellation in primely generated refinement monoids. Algebra Universalis 46, 343–371 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brown, L.G.: Homotopy of projections in C*-algebras of stable rank one. Recent Adv. Oper. Algebras (Orléans, 1992) 232, 115–120 (1995)Google Scholar
  11. 11.
    Brown, L.G., Pedersen, G.K.: C*-algebras of real rank zero. J. Funct. Anal. 99, 131–149 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brown, L.G., Pedersen, G.K.: Non-stable K-theory and extremally rich C*-algebras (Preprint)Google Scholar
  13. 13.
    Cuntz, J., Krieger, W.: A class of C*-algebras and topological Markov chains. Invent. Math. 56, 251–268 (1980)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Deicke, K., Hong, J.H., Szymański, W.: Stable rank of graph algebras. Type I graph algebras and their limits. Indiana Univ. Math. J. 52(4), 963–979 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Facchini, A., Halter-Koch, F.: Projective modules and divisor homomorphisms. J. Algebra Appl. 2, 435–449 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger, Malabar, Florida (1991)MATHGoogle Scholar
  17. 17.
    Jeong, J.A., Park, G.H.: Graph C*-algebras with real rank zero. J. Funct. Anal. 188, 216–226 (2002)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kumjian, A., Pask, D., Raeburn, I.: Cuntz–Krieger algebras of directed graphs. Pacific J. Math. 184, 161–174 (1998)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Leavitt, W.G.: The module type of a ring. Trans. Amer. Math. Soc. 42, 113–130 (1962)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Menal, P., Moncasi, J.: Lifting units in self-injective rings and an index theory for Rickart C*-algebras. Pacific J. Math. 126, 295–329 (1987)MATHMathSciNetGoogle Scholar
  21. 21.
    Perera, F.: Lifting units modulo exchange ideals and C*-algebras with real rank zero. J. Reine Angew. Math. 522, 51–62 (2000)MATHMathSciNetGoogle Scholar
  22. 22.
    Raeburn, I., Szymański, W.: Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc. 356, 39–59 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rieffel, M.A.: Dimension and stable rank in the K-theory of C*-algebras. Proc. London Math. Soc. 46, 301–333 (1983)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rørdam, M., Larsen, F., Laustsen, N.J.: An introduction to K-theory for C*-algebras. In: London Mathematical Society of Student Texts, vol. 49. Cambridge University Press, UK (2000)Google Scholar
  25. 25.
    Rosenberg, J.: Algebraic K-Theory and its Applications. Springer, Berlin Heidelberg New York, GTM 147 (1994)Google Scholar
  26. 26.
    Watatani, Y.: Graph theory for C*-algebras. In: Kadison, R.V. (ed.) Operator Algebras and Their Applications. Proceedings of Symposia in Pure Mathematics, vol. 38, Part I, pp. 195–197. Amer. Math. Soc. Providence, Rhode Island (1982)Google Scholar
  27. 27.
    Wehrung, F.: The dimension monoid of a lattice. Algebra Universalis 40, 247–411 (1998)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhang, S.: Diagonalizing projections in multiplier algebras and in matrices over a C*-algebra. Pacific J. Math. 54, 181–200 (1990)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Departamento de MatemáticasUniversidad de CádizPuerto RealSpain

Personalised recommendations