Baer and Baer *-ring characterizations of Leavitt path algebras

We characterize Leavitt path algebras which are Rickart, Baer, and Baer*-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer *-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well. Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer *-ring, a Rickart *-ring which is not Baer, or a Baer and not a Rickart *-ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their C*-algebra counterparts. For example, while a graph C*-algebra is Baer (and a Baer *-ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer *-ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops. (C) 2017 Elsevier B.V. All rights reserved.