# *-Regular Leavitt path algebras of arbitrary graphs

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10114-011-0106-8

- Cite this article as:
- Aranda Pino, G., Rangaswamy, K. & Vaš, L. Acta. Math. Sin.-English Ser. (2012) 28: 957. doi:10.1007/s10114-011-0106-8

## 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.