*-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 $^\ast$-regular (i.e. regular with proper involution). This characterization of $^\ast$-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 $^\ast$-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.