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Spectral Triples on O N

Part of the MATRIX Book Series book series (MXBS,volume 1)

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

  1. Connes, A.: Compact metric spaces, Fredholm modules and hyperfiniteness. Ergodic Theory Dyn. Syst. 9, 207–230 (1989)

    MATH  Google Scholar 

  2. Connes, A.: Noncommutative Geometry. Academic Press, London (1994)

    MATH  Google Scholar 

  3. 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)

    Google Scholar 

  4. 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)

    MathSciNet  CrossRef  Google Scholar 

  5. 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)

    Google Scholar 

  6. Cuntz, J.: Simple C -algebras generated by isometries. Commun. Math. Phys. 57(2), 173–185 (1977)

    MathSciNet  CrossRef  Google Scholar 

  7. Emerson, H., Nica, B.: K-homological finiteness and hyperbolic groups. J. Reine Angew. Math. (to appear). https://doi.org/10.1515/crelle-2015-0115

  8. Evans, D.E.: On O n . Publ. Res. Inst. Math. Sci. 16(3), 915–927 (1980)

    MathSciNet  CrossRef  Google Scholar 

  9. Farsi, C., Gillaspy, E., Julien, A., Kang, S., Packer, J.: Wavelets and spectral triples for fractal representations of Cuntz algebras. arXiv:1603.06979

    Google Scholar 

  10. Fröhlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and differential geometry. Commun. Math. Phys. 193(3), 527–594 (1998)

    MathSciNet  CrossRef  Google Scholar 

  11. Goffeng, M.: Equivariant extensions of ∗-algebras. N. Y. J. Math. 16, 369–385 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Goffeng, M., Mesland, B.: Spectral triples and finite summability on Cuntz-Krieger algebras. Doc. Math. 20, 89–170 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 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

  14. Kaminker, J., Putnam, I.: K-theoretic duality of shifts of finite type. Commun. Math. Phys. 187(3), 509–522 (1997)

    MathSciNet  CrossRef  Google Scholar 

  15. Kasparov, G.G.: The operator K-functor and extensions of C -algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44(3), 571–636, 719 (1980)

    Google Scholar 

  16. Laca, M., Neshveyev, S.: KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211, 457–482 (2004)

    MathSciNet  CrossRef  Google Scholar 

  17. Lord, S., Sukochev, F., Zanin, D.: Singular Traces. Theory and Applications. De Gruyter Studies in Mathematics, vol. 46. De Gruyter, Berlin (2013)

    Google Scholar 

  18. Olesen, D., Pedersen, G.K.: Some C -dynamical systems with a single KMS state. Math. Scand. 42, 111–118 (1978)

    MathSciNet  CrossRef  Google Scholar 

  19. Pask, D., Rennie, A.: The noncommutative geometry of graph C -algebras. I. The index theorem. J. Funct. Anal. 233(1), 92–134 (2006)

    MathSciNet  CrossRef  Google Scholar 

  20. Renault, J.: A Groupoid Approach to C -Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)

    CrossRef  Google Scholar 

  21. Renault, J.: Cuntz-like algebras. In: Operator Theoretical Methods (Timisoara, 1998), pp. 371–386. Theta Foundation, Bucharest (2000)

    Google Scholar 

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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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Correspondence to Magnus Goffeng .

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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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