Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 020, 21 pages      arXiv:2005.14225
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

A Spectral Triple for a Solenoid Based on the Sierpinski Gasket

Valeriano Aiello a, Daniele Guido b and Tommaso Isola b
a) Mathematisches Institut, Universität Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland
b) Dipartimento di Matematica, Università di Roma ''Tor Vergata'', I-00133 Roma, Italy

Received June 23, 2020, in final form February 10, 2021; Published online March 02, 2021

The Sierpinski gasket admits a locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed.

Key words: self-similar fractals; noncommutative geometry; ramified coverings.

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  1. Aiello V., Guido D., Isola T., Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448 (2017), 1378-1412, arXiv:1604.08619.
  2. Aiello V., Guido D., Isola T., Spectral triples on irreversible $C^*$-dynamical systems, arXiv:2102.05392.
  3. Arauza Rivera A., Spectral triples for the variants of the Sierpiński gasket, J. Fractal Geom. 6 (2019), 205-246, arXiv:1709.00755.
  4. Atiyah M.F., Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32-33 (1976), 43-72.
  5. Barlow M.T., Perkins E.A., Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), 543-623.
  6. Bellissard J.V., Marcolli M., Reihani K., Dynamical systems on spectral metric spaces, arXiv:1008.4617.
  7. Berestovskii V., Plaut C., Uniform universal covers of uniform spaces, Topology Appl. 154 (2007), 1748-1777, arXiv:math.AG/0607353.
  8. Carey A., Phillips J., Unbounded Fredholm modules and spectral flow, Canad. J. Math. 50 (1998), 673-718.
  9. Carey A., Phillips J., Sukochev F., Spectral flow and Dixmier traces, Adv. Math. 173 (2003), 68-113, arXiv:math.OA/0205076.
  10. Christensen E., Ivan C., Sums of two-dimensional spectral triples, Math. Scand. 100 (2007), 35-60, arXiv:math.OA/0601024.
  11. Christensen E., Ivan C., Lapidus M.L., Dirac operators and spectral triples for some fractal sets built on curves, Adv. Math. 217 (2008), 42-78, arXiv:math.MG/0610222.
  12. Christensen E., Ivan C., Schrohe E., Spectral triples and the geometry of fractals, J. Noncommut. Geom. 6 (2012), 249-274, arXiv:1002.3081.
  13. Cipriani F., Guido D., Isola T., A $C^*$-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and $L^2$-Betti numbers, J. Funct. Anal. 256 (2009), 603-634, arXiv:math.OA/0607603.
  14. Cipriani F., Guido D., Isola T., Sauvageot J.-L., Integrals and potentials of differential 1-forms on the Sierpinski gasket, Adv. Math. 239 (2013), 128-163, arXiv:1105.1995.
  15. Cipriani F., Guido D., Isola T., Sauvageot J.-L., Spectral triples for the Sierpinski gasket, J. Funct. Anal. 266 (2014), 4809-4869, arXiv:1112.6401.
  16. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  17. Cuntz J., The internal structure of simple $C^{\ast} $-algebras, in Operator Algebras and Applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., Vol. 38, Amer. Math. Soc., Providence, R.I., 1982, 85-115.
  18. Deeley R.J., Goffeng M., Mesland B., Whittaker M.F., Wieler solenoids, Cuntz-Pimsner algebras and $K$-theory, Ergodic Theory Dynam. Systems 38 (2018), 2942-2988, arXiv:1606.05449.
  19. Dixmier J., Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107-A1108.
  20. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, arXiv:hep-th/0303037.
  21. Fack T., Sur la notion de valeur caractéristique, J. Operator Theory 7 (1982), 307-333.
  22. Fack T., Kosaki H., Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), 269-300.
  23. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  24. Guido D., Isola T., Singular traces on semifinite von Neumann algebras, J. Funct. Anal. 134 (1995), 451-485.
  25. Guido D., Isola T., Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds, J. Funct. Anal. 176 (2000), 115-152, arXiv:math.OA/9802015.
  26. Guido D., Isola T., A semicontinuous trace for almost local operators on an open manifold, Internat. J. Math. 12 (2001), 1087-1102, arXiv:math.DG/0110294.
  27. Guido D., Isola T., Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal. 203 (2003), 362-400, arXiv:math.OA/0202108.
  28. Guido D., Isola T., Dimensions and spectral triples for fractals in ${\mathbb R}^N$, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 5, Theta, Bucharest, 2005, 89-108, arXiv:math.OA/0404295.
  29. Guido D., Isola T., Spectral triples for nested fractals, J. Noncommut. Geom. 11 (2017), 1413-1436, arXiv:1601.08208.
  30. Hawkins A., Skalski A., White S., Zacharias J., On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113 (2013), 262-291, arXiv:1103.6199.
  31. Higson N., Roe J., Analytic $K$-homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  32. Kigami J., Analysis on fractals, Cambridge Tracts in Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2001.
  33. Kigami J., Lapidus M.L., Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 217 (2001), 165-180.
  34. Lapidus M., Sarhad J., Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets, J. Noncommut. Geom. 8 (2014), 947-985, arXiv:1212.0878.
  35. Lapidus M.L., Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions, Topol. Methods Nonlinear Anal. 4 (1994), 137-195.
  36. Lapidus M.L., Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals, in Harmonic Analysis and Nonlinear Differential Equations (Riverside, CA, 1995), Contemp. Math., Vol. 208, Amer. Math. Soc., Providence, RI, 1997, 211-252.
  37. Latrémolière F., Packer J., Noncommutative solenoids and the Gromov-Hausdorff propinquity, Proc. Amer. Math. Soc. 145 (2017), 2043-2057, arXiv:1601.02707.
  38. McCord M.C., Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197-209.
  39. Nekrashevych V., Self-similar groups, Mathematical Surveys and Monographs, Vol. 117, Amer. Math. Soc., Providence, RI, 2005.
  40. Paterson A.L.T., Contractive spectral triples for crossed products, Math. Scand. 114 (2014), 275-298, arXiv:1204.4404.
  41. Rieffel M.A., Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215-229, arXiv:math.OA/9807084.
  42. Roe J., An index theorem on open manifolds. I, J. Differential Geom. 27 (1988), 87-113.
  43. Roe J., An index theorem on open manifolds. II, J. Differential Geom. 27 (1988), 115-136.
  44. Roe J., Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, Vol. 90, Amer. Math. Soc., Providence, RI, 1996.
  45. Roe J., Lectures on coarse geometry, University Lecture Series, Vol. 31, Amer. Math. Soc., Providence, RI, 2003.
  46. Ruan H.-J., Strichartz R.S., Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math. 61 (2009), 1151-1181.
  47. Sierpiński R.S., Sur une courbe dont tout point est un point de ramification, C. R. Acad. Sci. Paris 160 (1915), 302-305.
  48. Strichartz R.S., Fractals in the large, Canad. J. Math. 50 (1998), 638-657.
  49. Strichartz R.S., Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 4019-4043.
  50. Strichartz R.S., Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math. 61 (2009), 1182-1200.
  51. Teplyaev A., Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal. 159 (1998), 537-567.
  52. Willett R., Yu G., Higher index theory, Cambridge Studies in Advanced Mathematics, Vol. 189, Cambridge University Press, Cambridge, 2020.

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