Every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra

Abstract

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path algebra is graded isomorphic to $I$. Our proof adapts to show that for every closed gauge-invariant ideal $J$ of a graph $C^*$-algebra, there is a gauge-invariant $*$-isomorphism mapping the graph $C^*$-algebra of the porcupine graph of $J$ onto $J.$