Abstract
We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.
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Notes
This is different from Boava et al. (2017a). The authors realized that the original description of \(\mathscr {L}^{\infty }\) was incorrect—for instance, (Boava et al. 2017a, Proposition 4.18) did not hold with \(\mathscr {L}^{\infty }\) as originally described. With this change, all results involving \(\mathscr {L}^{\infty }\) hold.
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Boava, G., de Castro, G.G. & Mortari, F.d.L. Groupoid Models for the C*-Algebra of Labelled Spaces. Bull Braz Math Soc, New Series 51, 835–861 (2020). https://doi.org/10.1007/s00574-019-00177-6
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DOI: https://doi.org/10.1007/s00574-019-00177-6