Regularity conditions for arbitrary Leavitt path algebras

TL;DR

This paper characterizes when Leavitt path algebras are regular by establishing their structure for acyclic graphs and proving the equivalence of several regularity conditions.

Contribution

It provides a comprehensive set of equivalent conditions for regularity in Leavitt path algebras, linking algebraic properties to graph acyclicity.

Findings

Leavitt path algebra is locally K-matricial for acyclic graphs.

Regularity conditions are equivalent to acyclicity of the graph.

Additional regularity properties like unit regularity are also characterized.

Abstract

We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $\pi$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $\pi$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.