*-Regular Leavitt path algebras of arbitrary graphs

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.