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Noetherian Leavitt Path Algebras and Their Regular Algebras

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Abstract

In the past, it has been shown that the Leavitt path algebra L(E) = L K (E) of a graph E over a field K is left and right noetherian if and only if the graph E is finite and no cycle of E has an exit. If Q(E) = Q K (E) denotes the regular algebra over L(E), we prove that these two conditions are further equivalent with any of the following: L(E) contains no infinite set of orthogonal idempotents, L(E) has finite uniform dimension, L(E) is directly finite, Q(E) is directly finite, Q(E) is unit-regular, Q(E) is left (right) self-injective and a few more equivalences. In addition, if the involution on the field K is positive definite, these conditions are equivalent with the following: the involution * extends from L(E) to Q(E), Q(E) is *-regular, Q(E) is finite, Q(E) is the maximal (total or classical) symmetric ring of quotients of L(E), the maximal right ring of quotients of L(E) is the same as the total (or classical) left ring of quotients of L(E), every finitely generated nonsingular L(E)-module is projective, and the matrix ring M n (L(E)) is strongly Baer for every n. It may not be surprising that a noetherian Leavitt path algebra has these properties, but a more interesting fact is that these properties hold only if a Leavitt path algebra is noetherian (i.e. E is a finite no-exit graph).

Using some of these equivalences, we give a specific description of the inverse of the isomorphism V (L(E)) → V (Q(E)) of monoids of equivalence classes of finitely generated projective modules of L(E) and Q(E) for noetherian Leavitt path algebras. We also prove that two noetherian Leavitt path algebras are isomorphic as rings if and only if they are isomorphic as *-algebras. This answers in affirmative the Isomorphism Conjecture for the class of noetherian Leavitt path algebras: if \({L_{\mathbb{C}}(E)}\) and \({L_{\mathbb{C}}(F)}\) are noetherian Leavitt path algebras, then \({L_{\mathbb{C}}(E) \cong L_{\mathbb{C}}(F)}\) as rings implies \({C^{*} (E) \cong C^{*} (F)}\) as *-algebras.

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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  MathSciNet  MATH  Google Scholar 

  2. Abrams G., Aranda Pino G., Perera F., Siles Molina M.: Chain conditions for Leavitt path algebras. Forum Math. 22, 95–114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abrams G., Aranda Pino G., Siles Molina M.: Locally finite Leavitt path algebras. Israel J. Math. 165, 329–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abrams G., Rangaswamy K.L.: Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory 13(3), 319–334 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abrams G., Tomforde M.: Isomorphism and Morita equivalence of graph algebras. Trans. Amer. Math. Soc. 363, 3733–3767 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ara P., Brustenga M.: The regular algebra of a quiver. J. Algebra 309, 207–235 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ara P., Brustenga M.: Module theory over Leavitt path algebras and Ktheory. J. Pure Appl. Algebra 214(7), 1131–1151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ara P., Menal P.: On regular rings with involution. Arch. Math. (Basel) 42(2), 126–130 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ara P., Moreno M. A., Pardo E.: Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10, 157–178 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Aranda Pino, F. Perera and M. Siles Molina, eds., Graph algebras: bridging the gap between analysis and algebra, Universidad de Málaga Press 2007, ISBN: 978-84-9747-177-0.

  11. G. Aranda Pino, K. L. Rangaswamy and L. Vaš, *-regular Leavitt path algebra of arbitrary graphs. Acta Math. Sci. Ser. B Engl. Ed. 28 (5) (2012), 957 – 968.

  12. S. K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

  13. Bergman G. M.: Coproducts and Some Universal Ring Constructions. Trans. Amer. Math. Soc. 200, 33–88 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. W. Evans, A class of semihereditary rings. Rings, modules and radicals, (Hobart, 1987), 51–60, Pitman Res. Notes Math. Ser. 204, Longman Sci. Tech., Harlow, 1989.

  15. Gardner L. T.: On isomorphisms of C *-algebras, Amer. J. Math. 87((2), 384–396 (1965)

    Article  MATH  Google Scholar 

  16. K. R. Goodearl, Von Neumann Regular Rings, 2nd Ed., Krieger, Malabar, FL, 1991.

  17. Goodearl K.R.: Leavitt path algebras and direct limits. Contemp. Math. 480, 165–187 (2009)

    Article  MathSciNet  Google Scholar 

  18. Handelman D.: Perspectivity and cancellation in regular rings. J. Algebra 48, 1–16 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Handelman D.: Coordinatization applied to finite Baer *-rings. Trans. Amer. Math. Soc. 235, 1–34 (1978)

    MathSciNet  MATH  Google Scholar 

  20. Lanning S.: The maximal symmetric ring of quotients. J. Algebra, 179, 47–91 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1999.

  22. E. Ortega, The maximal symmetric ring of quotients: path algebras, incidence algebras and bicategories, Ph. D. Thesis, Universitat Autònoma de Barcelona, 2006.

  23. Ortega E.: Rings of quotients of incidence algebras and path algebras. J. Algebra 303(1), 225–243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ortega E.: Two-sided localization of bimodules, Comm. Algebra 36(5), 1911–1926 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pyle E.S.: The regular ring and the maximal ring of quotients of a finite Baer *-ring. Trans. Amer. Math. Soc. 203, 201–213 (1975)

    MathSciNet  MATH  Google Scholar 

  26. Siles Molina M.: Algebras of quotients of path algebras. J. Algebra 319(12), 329–348 (2008)

    Article  MathSciNet  Google Scholar 

  27. B. Stenström, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften 217, Springer-Verlag, New York-Heidelberg, 1975.

  28. Tomforde M.: Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra 318, 270–299 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Utumi Y.: On rings of which one-sided quotient rings are two-sided. Proc. Amer. Math. Soc. 14, 141–147 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vaš L.: Dimension and torsion theories for a class of Baer *-rings. J. Algebra 289(2), 614–639 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vaš L.: A simplification of Morita’s construction of total right rings of quotients for a class of rings. J. Algebra 304(2), 989–1003 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vaš L.: Perfect symmetric rings of quotients. J. Algebra Appl. 8(5), 689–711 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Gonzalo Aranda Pino

  2. Department of Mathematics, Physics and Statistics, University of the Sciences in Philadelphia, 600 S. 43rd St., Philadelphia, PA, 19104, USA

    Lia Vaš

Authors
  1. Gonzalo Aranda Pino
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  2. Lia Vaš
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Correspondence to Lia Vaš.

Additional information

The first author was partially supported by the Spanish MEC and Fondos FEDER through project MTM2007-60333, and by the Junta de Andalucía and Fondos FEDER, jointly, through projects FQM-336 and FQM-2467. Part of this work was carried out during a visit of the second author to the University of Málaga, partially funded by a “Grant for foreign visiting professors” within the III Research Framework Program of the University of Málaga. The second author thanks this host institution for its warm hospitality and support.

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Aranda Pino, G., Vaš, L. Noetherian Leavitt Path Algebras and Their Regular Algebras. Mediterr. J. Math. 10, 1633–1656 (2013). https://doi.org/10.1007/s00009-013-0320-y

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  • DOI: https://doi.org/10.1007/s00009-013-0320-y

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