Acta Mathematica Sinica, English Series

, Volume 28, Issue 5, pp 957–968 | Cite as

*-Regular Leavitt path algebras of arbitrary graphs

  • Gonzalo Aranda Pino
  • Kulumani Rangaswamy
  • Lia Vaš
Article

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 LK(E). We show that the involution on LK(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 LK(E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK(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.

Keywords

Leavitt path algebra *-regular involution arbitrary graph 

MR(2000) Subject Classification

16D70 16W10 16S99 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Gonzalo Aranda Pino
    • 1
  • Kulumani Rangaswamy
    • 2
  • Lia Vaš
    • 3
  1. 1.Departamento de Álgebra, Geometría y TopologíaUniversidad de MálagaMálagaSpain
  2. 2.Department of MathematicsUniversity of ColoradoColorado SpringsUSA
  3. 3.Department of Mathematics, Physics and StatisticsUniversity of the Sciences in PhiladelphiaPhiladelphiaUSA

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