Abstract
Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.
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The first author is supported by the Marsden grant 15-UOO-071 from the Royal Society of New Zealand. The last three authors are supported by the Junta de Andalucía and Fondos FEDER, jointly, through projects FQM-336 and FQM-7156. They are also supported by the Spanish Ministerio de Economía y Competitividad and Fondos FEDER, jointly, through project MTM2016-76327-C3-1-P. This research took place while the first author was visiting the Universidad de Málaga. She thanks her coauthors for their hospitality. The authors also thank the anonymous referee for their helpful comments.
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Clark, L.O., Martín Barquero, D., Martín González, C. et al. Using the Steinberg algebra model to determine the center of any Leavitt path algebra. Isr. J. Math. 230, 23–44 (2019). https://doi.org/10.1007/s11856-018-1816-8
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DOI: https://doi.org/10.1007/s11856-018-1816-8