Regular ideals of graph algebras

TL;DR

This paper characterizes the structure of regular ideals in graph C*-algebras, showing how they relate to gauge-invariance and preserving Condition (L) in quotients, with a complete description of their vertex sets.

Contribution

It provides a full description of gauge-invariant regular ideals in graph C*-algebras and demonstrates their stability under quotients when Condition (L) is satisfied.

Findings

Regular ideals are gauge-invariant under Condition (L).

Quotients by regular ideals preserve Condition (L).

Vertex sets of regular ideals are fully characterized.

Abstract

Let $C^*(E)$ be the graph C$^*$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^*(E)$. It is shown that when $E$ satisfies Condition (L) the regular ideals $C^*(E)$ are a class of gauge-invariant ideals which preserve Condition (L) under quotients. That is, we show that if $E$ satisfies Condition (L) then a regular ideal $J \unlhd C^*(E)$ is necessarily gauge-invariant. Further, if $J \unlhd C^*(E)$ is a regular ideal, it is shown that $C^*(E)/J \simeq C^*(F)$ where $F$ satisfies Condition (L).