Baer and Baer *-ring characterizations of Leavitt path algebras

Abstract

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.

Graded Chain Conditions and Leavitt Path Algebras of No-Exit Graphs

“…(i) Results of [14] and [16] imply that the following conditions are equivalent with conditions from Theorem 3.4. …”

“…Assuming (4'), it follows that L K (E) is a locally Baer and regular ring. By [16,Theorem 15], E is a row-finite, no-exit graph and every infinite path ends in a sink or a cycle. The regularity of L K (E) implies that E is acyclic by [2, Theorem 1] so (5') holds.…”

“…We characterize the graded and the non-graded artinian-related properties of Leavitt path algebras without imposing any restrictions on the cardinality of the graph. We let (16) and (17) stand for statements (16l) and (17l) respectively, with the word "left" deleted.…”

A Survey of Some of the Recent Developments in Leavitt Path Algebras

“…Theorem 4.7 ( [9], [17], [25], [27], [28], [43] ) For a finite graph E, the following conditions are equivalent for L := L K (E):…”

The Groupoid Approach to Leavitt Path Algebras

“…This is useful for generating counterexamples to reasonable-sounding conjectures, e.g. [6,46], or for supporting other long-standing conjectures by showing they hold within this varied class, e.g. [9,15].…”