Abstract
If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L K (E). We show that the involution on L K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.
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References
Cuntz, J.: Simple C*-algebras generated by isometries. Comm. Math. Phys., 57, 173–185 (1977)
Leavitt, W. G.: Modules without invariant basis number. Proc. Amer. Math. Soc., 8, 322–328 (1957)
Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra, 293(2), 319–334 (2005)
Ara, P., Moreno, M. A., Pardo, E.: Nonstable K-theory for graph algebras. Algebra Represent. Theory, 10, 157–178 (2007)
Aranda Pino, G., Pardo, E., Siles Molina, M.: Exchange Leavitt path algebras and stable rank. J. Algebra, 305(2), 912–936 (2006)
Aranda Pino, G., Martín Barquero, D., Martín González, C., et al.: The socle of a Leavitt path algebra. J. Pure Appl. Algebra, 212(3), 500–509 (2008)
Abrams, G., Tomforde, M.: Isomorphism and Morita equivalence of graph algebras. Trans. Amer. Math. Soc., 363(7), 3733–3767 (2011)
Abrams, G., ánh, P. N., Louly, A., et al.: The classification question for Leavitt algebras. J. Algebra, 320, 1983–2026 (2008)
Ara, P., Brustenga, M., Cortiñas, G.: K-theory for Leavitt path algebras. Münster J. Math., 2, 5–33 (2009)
Tomforde, M.: Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra, 318(1), 270–299 (2007)
Goodearl, K. R.: Leavitt path algebras and direct limits. Contemp. Math., 480, 165–187 (2009)
Raeburn, I.: Chapter in “Graph Algebras: Bridging the Gap between Analysis and Algebra” (G. Aranda Pino, F. Perera, M. Siles Molina, eds.), ISBN: 978-84-9747-177-0, University of Málaga Press, Málaga, Spain, 2007
Abrams, G., Rangaswamy, K. M.: Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory, 13, 319–334 (2010)
Siles Molina, M.: Algebras of quotients of Leavitt path algebras. J. Algebra, 319(12), 5265–5278 (2008)
Abrams, G., Aranda Pino, G.: Purely infinite simple Leavitt path algebras. J. Pure Appl. Algebra, 207(3), 553–563 (2006)
Drinen, D., Tomforde, M.: The C*-algebras of arbitrary graphs. Rocky Mountain J. Math., 35(1), 105–135 (2005)
Abrams, G., Aranda Pino, G.: The Leavitt path algebras of arbitrary graphs. Houston J. Math., 34(2), 423–442 (2008)
Lam, T. Y.: A First Course on Noncommutative Rings, Springer-Verlag, New York, 1991
Berberian, S. K.: Baer *-Rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972
Abrams, G., Aranda Pino, G., Siles Molina, M.: Finite-dimensional Leavitt path algebras. J. Pure Appl. Algebra, 209(3), 753–762 (2007)
Goodearl, K. R.: Von Neumann Regular Rings, Second Ed., Krieger, Malabar, FL, 1991
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The first author is 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
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Aranda Pino, G., Rangaswamy, K. & Vaš, L. *-Regular Leavitt path algebras of arbitrary graphs. Acta. Math. Sin.-English Ser. 28, 957–968 (2012). https://doi.org/10.1007/s10114-011-0106-8
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DOI: https://doi.org/10.1007/s10114-011-0106-8