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Strong gradings on Leavitt path algebras, Steinberg algebras and their \(C^*\)-completions

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Abstract

We show the reduced \(C^*\)-algebra of a graded ample groupoid is a strongly graded \(C^*\)-algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We use this to show that the graph \(C^*\)-algebra of a countable directed graph is a strongly \({\mathbb {Z}}\)-graded \(C^*\)-algebra if and only if the Leavitt path algebra is a strongly \({\mathbb {Z}}\)-graded ring. The latter of these was shown previously to be equivalent to the graph satisfying property (Y).

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Notes

  1. Paterson calls \(G_E\) the path groupoid

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington, 6140, New Zealand

    Lisa Orloff Clark & Ellis Dawson

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  1. Lisa Orloff Clark
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  2. Ellis Dawson
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Correspondence to Lisa Orloff Clark.

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This research was supported by a Marsden grant from the Royal Society of New Zealand. The authors thank Iain Raeburn for sharing his insights. They also thank the anonymous referees for their helpful suggestions.

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Clark, L.O., Dawson, E. Strong gradings on Leavitt path algebras, Steinberg algebras and their \(C^*\)-completions. J Algebr Comb 58, 453–464 (2023). https://doi.org/10.1007/s10801-022-01191-6

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  • DOI: https://doi.org/10.1007/s10801-022-01191-6

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