Baer and Baer *-ring characterizations of Leavitt path algebras

TL;DR

This paper characterizes Leavitt path algebras as Rickart, Baer, and Baer *-rings based on graph properties, extending to non-unital and graded cases, and compares these with $C^*$-algebra counterparts.

Contribution

It provides new graph-theoretic characterizations of Leavitt path algebras as various annihilator-related rings, including non-unital and graded versions, enabling easy generation of examples.

Findings

Leavitt path algebra is Baer iff the graph is finite with no cycles with exits.

Leavitt path algebra is Baer *-ring iff the graph is a finite disjoint union of finite acyclic graphs or loops.

Characterizations differ from $C^*$-algebras, e.g., in conditions for being Baer or Baer *-ring.

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.