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EVERY GRADED IDEAL OF A LEAVITT PATH ALGEBRA IS GRADED ISOMORPHIC TO A LEAVITT PATH ALGEBRA

Published online by Cambridge University Press:  23 August 2021

LIA VAŠ*
Affiliation:
Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA19104, USA

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 generalised 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 generalised hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) whose Leavitt path algebra is graded isomorphic to I. Our proof can be adapted 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 $J.$

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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