Baer and Baer *-ring characterizations of Leavitt path algebras

Abstract

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.