Baer and Baer *-ring characterizations of Leavitt path algebras
Roozbeh Hazrat, Lia Vas
TL;DR
The paper addresses the problem of characterizing Leavitt path algebras $L_K(E)$ that are Baer, Baer $*$-rings, and Rickart (including locally unital and graded variants) in terms of the underlying graph and the base field. It develops annihilator-related notions for graded and locally unital rings, extends them to corners, and establishes graph-theoretic criteria, including no-exit and finiteness conditions, that determine when LPAs possess these properties; it also analyzes positivity requirements of the base field and provides matrix-ring decompositions in key cases. The main contributions include precise graph criteria for Baer, Baer $*$-ring, and Rickart LPAs, their graded and local versions, and the demonstration that graded and local variants can differ from their ungraded, global counterparts, yielding a rich landscape distinct from graph $C^*$-algebras. The work enables systematic generation of rings with prescribed annihilator properties and clarifies the interplay between algebraic and graph structures, while also highlighting open questions and subtle distinctions in the Leavitt path algebra setting.
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.