A note on the regular ideals of Leavitt path algebras

TL;DR

This paper proves that regular ideals in Leavitt path algebras are graded, and explores their structure and properties, including how they relate to quotients and graph conditions, with comparisons to graph C*-algebras.

Contribution

It establishes that regular ideals are graded in Leavitt path algebras and analyzes their structure and implications for quotients and graph conditions.

Findings

Regular ideals are graded in Leavitt path algebras.

Quotients by regular ideals are again Leavitt path algebras for row-finite graphs.

Condition (L) is preserved under quotients by regular ideals.

Abstract

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition~(L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.