Abstract
We give a construction of an odd spectral triple on the Cuntz algebra O N , whose K-homology class generates the odd K-homology group K 1(O N ). Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on O N yields a θ-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on O N is discussed.
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Acknowledgements
We thank the MATRIX for the program Refining C ∗ -algebraic invariants for dynamics using KK-theory in Creswick, Australia (2016) where this work came into being. We are grateful to the support from Leibniz University Hannover where this work was initiated. We also thank Francesca Arici, Robin Deeley, Adam Rennie and Alexander Usachev for fruitful discussions and helpful comments, and the anonymous referee for a careful reading of the manuscript. The first author was supported by the Swedish Research Council Grant 2015-00137 and Marie Sklodowska Curie Actions, Cofund, Project INCA 600398.
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Goffeng, M., Mesland, B. (2018). Spectral Triples on O N . In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_9
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DOI: https://doi.org/10.1007/978-3-319-72299-3_9
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