Regular ideals of graph algebras

Let $C^{\ast }(E)$ be the graph C${}^{\ast }$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^{\ast }(E)$. It is shown that when $E$ satisfies Condition (L) the regular ideals $C^{\ast }(E)$ 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 $J⊴C^{\ast }(E)$ is necessarily gauge-invariant. Further, if $J⊴C^{\ast }(E)$ is a regular ideal, it is shown that $C^{\ast }(E)\mathrm{/}J\simeq C^{\ast }(F)$ where $F$ satisfies Condition (L).