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A REMARK ON GEL'FAND DUALITY FOR SPECTRAL TRIPLES

  • Bertozzini, Paolo (Department of Mathematics and Statistics Faculty of Science and Technology Thammsat University - Rangsit Campus) ;
  • Conti, Roberto (Mathematics School of Mathematical and Physical Sciences University of Newcastle) ;
  • Lewkeeratiyutkul, Wicharn (Department of Mathematics Faculty of Science Chulalongkorn University)
  • Received : 2009.09.10
  • Published : 2011.05.31

Abstract

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-$C^*$-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in [5] into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.

References

  1. J. Baez, Categories, quantization and much more, http://math.ucr.edu/home/baez/ categories.html, 26 September 2004.
  2. M. Barr and C. Wells, Category Theory for Computing Science, third edition, Centre de Recherches Mathematiques, Montreal, 1999.
  3. N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer- Verlag, Berlin, 1992.
  4. P. Bertozzini, Categories of spectral geometries, Seminar Slides and Video of the Talk at the "Categories Logic and Physics" workshop, Imperial College, London, 14 May 2008, http://categorieslogicphysics.wikidot.com/people#paolobertozzini.
  5. P. Bertozzini, R. Conti, and W. Lewkeeratiyutkul, A category of spectral triples and discrete groups with length function, Osaka J. Math. 43 (2006), no. 2, 327-350.
  6. P. Bertozzini, R. Conti, and W. Lewkeeratiyutkul, Non-commutative geometry, categories and quantum physics, East-West J. Math. 2007 (2007), Special Vol., 213-259.
  7. A. Connes, Noncommutative Geometry, Academic Press, 1994.
  8. A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995), no. 11, 6194-6231. https://doi.org/10.1063/1.531241
  9. A. Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), no. 1, 155-176. https://doi.org/10.1007/BF02506388
  10. A. Connes, Brisure de symetrie spontanee et geometrie du pont de vue spectral, J. Geom. Phys. 23 (1997), no. 3-4, 206-234. https://doi.org/10.1016/S0393-0440(97)80001-0
  11. A. Connes, On the spectral characterization of manifolds, arXiv:0810.2088.
  12. A. Connes,A unitary invariant in Riemannian geometry, Int. J. Geom. Methods Mod. Phys. 5 (2008), no. 8, 1215-1242. https://doi.org/10.1142/S0219887808003284
  13. I. Dolgachev, Book Review: "The Geometry of Schemes" by D. Eisenbud and J. Harris, Springer, 2000, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 4, 467-473. https://doi.org/10.1090/S0273-0979-01-00911-9
  14. S. Doplicher and J. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989), no. 1, 157-218. https://doi.org/10.1007/BF01388849
  15. I. Gel'fand, Normierte ringe, Mat. Sbornik 51 (1941), no. 9, 3-24.
  16. I. Gel'fand and M. Naimark, On the imbedding of normed rings into the ring of operators in Hilbert space, Math. Sbornik 12(54) (1943), 197-213.
  17. J.-M. Gracia-Bondia, H. Figueroa, and J.-C. Varilly, Elements of Noncommutative Geometry, Birkhauser, 2001.
  18. P. Halmos and J. von Neumann, Operator Methods in Classical Mechanics II, Ann. Math. 43 (1942), 332-350. https://doi.org/10.2307/1968872
  19. D. Hilbert, Uber die vollen Invariantesysteme, Math. Ann. 42 (1893), no. 3, 313-373. https://doi.org/10.1007/BF01444162
  20. M. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady Acad. Nauk SSSR (N. S.) 69 (1949), 725-728.
  21. H.-B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989.
  22. S. MacLane, Categories for the Working Mathematician, Springer, 1998.
  23. B. Mesland, Unbounded biviariant K-theory and correspondences in noncommutative geometry, arXiv:0904.4383v1.
  24. P. Petersen, Riemannian Geometry, Springer-Verlag, 1998.
  25. L. Pontryagin, The theory of topological commutative groups, Ann. Math. 35 (1934), no. 2, 361-388. https://doi.org/10.2307/1968438
  26. A. Rennie, Commutative geometries are spin manifolds, Rev. Math. Phys. 13 (2001), no. 4, 409-464. https://doi.org/10.1142/S0129055X01000673
  27. A. Rennie and J. Varilly, Reconstruction of manifolds in noncommutative geometry, arXiv:math/0610418.
  28. A. Rennie and J. Varilly, Orbifolds are not commutative geometries, J. Aust. Math. Soc. 84 (2008), no. 1, 109-116.
  29. M.-A. Rieffel, Compact quantum metric spaces, Operator algebras, quantization, and noncommutative geometry, 315-330, Contemp. Math., 365, Amer. Math. Soc., Providence, RI, 2004. https://doi.org/10.1090/conm/365/06709
  30. H. Schroder, On the definition of geometric Dirac operators, arXiv:math.DG/0005239.
  31. J.-P. Serre, Modules projectifs et espaces fibre a fibre vectorielle, expose 23, Seminaire Dubreil-Pisot, Paris, 1958.
  32. M. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), no. 1, 37-111.
  33. M. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375-481. https://doi.org/10.1090/S0002-9947-1937-1501905-7
  34. R. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277. https://doi.org/10.1090/S0002-9947-1962-0143225-6
  35. A. Takahashi, Hilbert modules and their representation, Rev. Colombiana Mat. 13 (1979), no. 1, 1-38.
  36. A. Takahashi, A duality between Hilbert modules and fields of Hilbert spaces, Rev. Colombiana Mat. 13 (1979), no. 2, 93-120.
  37. T. Tannaka, Uber den dualitatssatz der nichtkommutativen topologischen gruppen, Tohoku Math. J. 45 (1939), 1-12.
  38. E. van Kampen, Locally bicompact abelian groups and their character groups, Ann. Math. 36 (1935), no. 2, 448-463. https://doi.org/10.2307/1968582