, Volume 28, Issue 5, pp 957-968

*-Regular Leavitt path algebras of arbitrary graphs

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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.

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