*-Regular Leavitt path algebras of arbitrary graphs |
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Authors: | Email author" target="_blank">Gonzalo?Aranda PinoEmail author Kulumani?Rangaswamy Lia?Va? |
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Institution: | [1]Departamento de Algebra, Geometriay Topologia, Universidad de Mdlaga, 29071 Mdlaga, Spain [2]Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA [3]Department of Mathematics, Physics and Statistics, University of the Sciences in Philadelphia, Philadelphia, PA 19104, USA |
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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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Keywords: | Leavitt path algebra *-regular involution arbitrary graph |
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