2016/06/17 by Hazrat, Roozbeh, Vas, Lia
#16D70 #16S10 #16W10 #16W50 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1606.05692
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.