Higher localized analytic indices and strict deformation quantization

  • Paulo Carrillo Rouse


This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let G be a Lie groupoid with Lie algebroid AG. Let τ be a (periodic) cyclic cocycle over the convolution algebra \( C_c^\infty \left( G \right) \) We say that τ can be localized if there is a morphism
$$ K^0 \left( {A^* G} \right)\buildrel {Ind_\tau } \over \longrightarrow C $$
satisfying Ind τ (a)=〈ind D a, τ 〉 (Connes pairing). In this case, we call Ind τ the higher localized index associated to τ. In [CR08a] we use the algebra of functions over the tangent groupoid introduced in [CR08b], which is in fact a strict deformation quantization of the Schwartz algebra S(AG ), to prove the following results:
  • Every bounded continuous cyclic cocycle can be localized.

  • If G is étale, every cyclic cocycle can be localized.

We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting.


Vector Bundle Elliptic Operator Hyperbolic Group Principal Symbol Analytic Index 
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  1. J. Aastrup, S.T. Melo, B. Monthubert, and E. Schrohe, Boutet de monvel’s calculus and groupoids i.Google Scholar
  2. M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530.CrossRefMathSciNetGoogle Scholar
  3. Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of C∗-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363.MATHMathSciNetGoogle Scholar
  4. J.-L. Brylinski and Victor Nistor, Cyclic cohomology of étale groupoids, K-Theory 8 (1994), no. 4, 341–365.MATHCrossRefMathSciNetGoogle Scholar
  5. Dan Burghelea, The cyclic homology of the group rings, Comment. Math. Helv. 60 (1985), no. 3, 354–365.MATHCrossRefMathSciNetGoogle Scholar
  6. Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000.Google Scholar
  7. Alain Connes and Nigel Higson, Déformations, morphismes asymptotiques et K-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106.MATHMathSciNetGoogle Scholar
  8. Alain Connes and Henri Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), no. 3, 345–388.MATHCrossRefMathSciNetGoogle Scholar
  9. Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143.Google Scholar
  10. _____, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. (1985), no. 62, 257–360.Google Scholar
  11. _____, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 52–144.Google Scholar
  12. _____, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.MATHGoogle Scholar
  13. Paulo Carrillo-Rouse, Compactly supported analytic indices for lie groupoids, Accepted in Journal of K-theory. Arxiv:math.KT/0803.2060 (2008).Google Scholar
  14. _____, A Schwartz type algebra for the tangent groupoid, ”K-theory and Noncommutative Geometry” Book EMS, edited by G. Cortinas, J. Cuntz, M. Karoubi, R. Nest and C. Weibel (2008).Google Scholar
  15. Marius Crainic, Cyclic cohomology of étale groupoids: the general case, K-Theory 17 (1999), no. 4, 319–362.MATHCrossRefMathSciNetGoogle Scholar
  16. Alain Connes and Georges Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183.MATHCrossRefMathSciNetGoogle Scholar
  17. Joachim Cuntz, Georges Skandalis, and Boris Tsygan, Cyclic homology in noncommutative geometry, Encyclopaedia of Mathematical Sciences, vol. 121, Springer-Verlag, Berlin, 2004,, Operator Algebras and Non-commutative Geometry, II.Google Scholar
  18. Jacques Dixmier, Les C∗-algbres et leurs reprsentations, Gauthier-Villars.Google Scholar
  19. Claire Debord, Jean-Marie Lescure, and Victor Nistor, Groupoids and an index theorem for conical pseudomanifolds, arxiv:math.OA/0609438 (2006).Google Scholar
  20. Charles Ehresmann, Catégories et structures, Dunod, Paris, 1965.MATHGoogle Scholar
  21. Alexander Gorokhovsky and John Lott, Local index theory over étale groupoids, J. Reine Angew. Math. 560 (2003), 151–198.MATHMathSciNetGoogle Scholar
  22. _____, Local index theory over foliation groupoids, Adv. Math. 204 (2006), no. 2, 413–447.MATHCrossRefMathSciNetGoogle Scholar
  23. Claude Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991, Études géométriques. [Geometric studies], With a preface by G. Reeb.Google Scholar
  24. Thomas G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215.MATHCrossRefMathSciNetGoogle Scholar
  25. André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque (1984), no. 116, 70–97, Transversal structure of foliations (Toulouse, 1982).Google Scholar
  26. Michel Hilsum and Georges Skandalis, Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 325–390.MATHGoogle Scholar
  27. Max Karoubi, Homologie cyclique et K-théorie, Astérisque (1987), no. 149, 147.Google Scholar
  28. Eric Leichtnam and Paolo Piazza, Étale groupoids, eta invariants and index theory, J. Reine Angew. Math. 587 (2005), 169–233.MATHMathSciNetGoogle Scholar
  29. K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, 1987.Google Scholar
  30. Ieke Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205–222.Google Scholar
  31. Bertrand Monthubert, Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal. 199 (2003), no. 1, 243–286.MATHCrossRefMathSciNetGoogle Scholar
  32. Bertrand Monthubert and François Pierrot, Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 2, 193–198.MATHMathSciNetGoogle Scholar
  33. H. Moscovici and F.-B. Wu, Localization of topological Pontryagin classes via finite propagation speed, Geom. Funct. Anal. 4 (1994), no. 1, 52–92.MATHCrossRefMathSciNetGoogle Scholar
  34. Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152.MATHCrossRefMathSciNetGoogle Scholar
  35. Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston Inc., Boston, MA, 1999.Google Scholar
  36. Jean Renault, A groupoid approach to C∗-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.Google Scholar
  37. François Trèves, Topological vector spaces, distributions and kernels, Dover Publications Inc., Mineola, NY, 2006, Unabridged republication of the 1967 original.Google Scholar
  38. H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75.MATHCrossRefMathSciNetGoogle Scholar

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  • Paulo Carrillo Rouse

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