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Nonstable K-theory for Graph Algebras

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Abstract

We compute the monoid V(L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (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 L K (E) and the lattice of order-ideals of V(L K (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.

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References

  1. Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ara, P., Facchini, A.: Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math. 18, 365–389 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bates, T., Pask, D., Raeburn, I., Szymański, W.: The C*-algebras of row-finite graphs. New York J. Math. 6, 307–324 (2000)

    MATH  MathSciNet  Google Scholar 

  6. Bergman, G.M.: Coproducts and some universal ring constructions. Trans. Amer. Math. Soc. 200, 33–88 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Blackadar, B.: Rational C*-algebras and nonstable K-theory. Rocky Mountain J. Math. 20(2), 285–316 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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. Brookfield, G.: Cancellation in primely generated refinement monoids. Algebra Universalis 46, 343–371 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Brown, L.G., Pedersen, G.K.: C*-algebras of real rank zero. J. Funct. Anal. 99, 131–149 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Brown, L.G., Pedersen, G.K.: Non-stable K-theory and extremally rich C*-algebras (Preprint)

  13. Cuntz, J., Krieger, W.: A class of C*-algebras and topological Markov chains. Invent. Math. 56, 251–268 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Facchini, A., Halter-Koch, F.: Projective modules and divisor homomorphisms. J. Algebra Appl. 2, 435–449 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger, Malabar, Florida (1991)

    MATH  Google Scholar 

  17. Jeong, J.A., Park, G.H.: Graph C*-algebras with real rank zero. J. Funct. Anal. 188, 216–226 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kumjian, A., Pask, D., Raeburn, I.: Cuntz–Krieger algebras of directed graphs. Pacific J. Math. 184, 161–174 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Leavitt, W.G.: The module type of a ring. Trans. Amer. Math. Soc. 42, 113–130 (1962)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  21. Perera, F.: Lifting units modulo exchange ideals and C*-algebras with real rank zero. J. Reine Angew. Math. 522, 51–62 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Raeburn, I., Szymański, W.: Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc. 356, 39–59 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rieffel, M.A.: Dimension and stable rank in the K-theory of C*-algebras. Proc. London Math. Soc. 46, 301–333 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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. Rosenberg, J.: Algebraic K-Theory and its Applications. Springer, Berlin Heidelberg New York, GTM 147 (1994)

    Google Scholar 

  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)

  27. Wehrung, F.: The dimension monoid of a lattice. Algebra Universalis 40, 247–411 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang, S.: Diagonalizing projections in multiplier algebras and in matrices over a C*-algebra. Pacific J. Math. 54, 181–200 (1990)

    Google Scholar 

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, (Barcelona), Spain

    P. Ara

  2. Departamento de Matemáticas, Universidad de Cádiz, Apartado 40, 11510, Puerto Real, (Cádiz), Spain

    M. A. Moreno & E. Pardo

Authors
  1. P. Ara
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  2. M. A. Moreno
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  3. E. Pardo
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Correspondence to P. Ara.

Additional information

The first author was partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM2002-01390, the second and the third by the DGI and European Regional Development Fund, jointly, through Project MTM2004-00149 and by PAI III grant FQM-298 of the Junta de Andalucía. Also, the first and third authors are partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.

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Ara, P., Moreno, M.A. & Pardo, E. Nonstable K-theory for Graph Algebras. Algebr Represent Theor 10, 157–178 (2007). https://doi.org/10.1007/s10468-006-9044-z

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