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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References
Connes, A.: Compact metric spaces, Fredholm modules and hyperfiniteness. Ergodic Theory Dyn. Syst. 9, 207–230 (1989)
Connes, A.: Noncommutative Geometry. Academic Press, London (1994)
Connes, A., Moscovici, H.: Type III and spectral triples. In: Traces in Number Theory, Geometry and Quantum Fields. Aspects of Mathematics, vol. E38, pp. 57–71. Friedrich Vieweg, Wiesbaden (2008)
Coornaert, M.: Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pac. J. Math. 159(2), 241–270 (1993)
Cornelissen, G., Marcolli, M., Reihani, K., Vdovina, A.: Noncommutative geometry on trees and buildings. In: Traces in Geometry, Number Theory, and Quantum Fields, pp. 73–98. Vieweg, Wiesbaden (2007)
Cuntz, J.: Simple C ∗-algebras generated by isometries. Commun. Math. Phys. 57(2), 173–185 (1977)
Emerson, H., Nica, B.: K-homological finiteness and hyperbolic groups. J. Reine Angew. Math. (to appear). https://doi.org/10.1515/crelle-2015-0115
Evans, D.E.: On O n . Publ. Res. Inst. Math. Sci. 16(3), 915–927 (1980)
Farsi, C., Gillaspy, E., Julien, A., Kang, S., Packer, J.: Wavelets and spectral triples for fractal representations of Cuntz algebras. arXiv:1603.06979
Fröhlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and differential geometry. Commun. Math. Phys. 193(3), 527–594 (1998)
Goffeng, M.: Equivariant extensions of ∗-algebras. N. Y. J. Math. 16, 369–385 (2010)
Goffeng, M., Mesland, B.: Spectral triples and finite summability on Cuntz-Krieger algebras. Doc. Math. 20, 89–170 (2015)
Goffeng, M., Mesland, B., Rennie, A.: Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension. Ergodic Theory Dyn. Syst. (to appear). https://doi.org/10.1017/etds.2016.75
Kaminker, J., Putnam, I.: K-theoretic duality of shifts of finite type. Commun. Math. Phys. 187(3), 509–522 (1997)
Kasparov, G.G.: The operator K-functor and extensions of C ∗-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44(3), 571–636, 719 (1980)
Laca, M., Neshveyev, S.: KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211, 457–482 (2004)
Lord, S., Sukochev, F., Zanin, D.: Singular Traces. Theory and Applications. De Gruyter Studies in Mathematics, vol. 46. De Gruyter, Berlin (2013)
Olesen, D., Pedersen, G.K.: Some C ∗-dynamical systems with a single KMS state. Math. Scand. 42, 111–118 (1978)
Pask, D., Rennie, A.: The noncommutative geometry of graph C ∗-algebras. I. The index theorem. J. Funct. Anal. 233(1), 92–134 (2006)
Renault, J.: A Groupoid Approach to C ∗-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Renault, J.: Cuntz-like algebras. In: Operator Theoretical Methods (Timisoara, 1998), pp. 371–386. Theta Foundation, Bucharest (2000)
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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