*-Regular Leavitt Path Algebras of Arbitrary Graphs

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 graph- theoretic 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.