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Lie groupoids, pseudodifferential calculus, and index theory

  • Claire Debord
  • Georges SkandalisEmail author
Chapter

Abstract

Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C-algebras, their pseudodifferential calculus, etc. We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.

Notes

Acknowledgements

The authors were partially supported by ANR-14-CE25-0012-01 (SINGSTAR).

References

  1. 1.
    Johannes Aastrup, Severino T. Melo, Bertrand Monthubert, and Elmar Schrohe, Boutet de Monvel’s calculus and groupoids I., J. Noncommut. Geom. 4 no. 3 (2010), 313–329.Google Scholar
  2. 2.
    Rui Almeida and Pierre Molino, Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 1, 13–15. MR 778785Google Scholar
  3. 3.
    _________ , Flots riemanniens sur les 4-variétés compactes, Tohoku Math. J. (2) 38 (1986), no. 2, 313–326. MR 843815MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000, With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683Google Scholar
  5. 5.
    Iakovos Androulidakis and Georges Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math. 626 (2009), 1–37.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    _________ , Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom. 5 (2011), no. 1, 125–152.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Michael F. Atiyah, Elliptic operators discrete groups and von Neumann algebras, Astérisque 32–33 (1976), 43–72.zbMATHGoogle Scholar
  8. 8.
    Michael F. Atiyah and Singer Isadore M., The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530.Google Scholar
  9. 9.
    _________ , The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 0236952Google Scholar
  10. 10.
    _________ , The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138.Google Scholar
  11. 11.
    Paul Baum and Alain Connes, GeometricK-theory for Lie groups and foliations, Enseign. Math. (2) 46 (2000), no. 1-2, 3–42. MR 1769535Google Scholar
  12. 12.
    Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions andK-theory of groupC -algebras, C -algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018Google Scholar
  13. 13.
    Paul Baum, Erik Guentner, and Rufus Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, Ann. K-Theory 1 (2016), no. 2, 155–208. MR 3514939MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Richard Beals and Peter Greiner, Calculus on Heisenberg manifolds, Annals of Mathematics Studies, vol. 119, Princeton University Press, Princeton, NJ, 1988. MR 953082Google Scholar
  15. 15.
    Moulay Tahar Benameur and Indrava Roy, The Higson-Roe sequence for étale groupoids. I. dual algebras and compatibility with the BC map, arXiv:1801.06040.Google Scholar
  16. 16.
    Karsten Bohlen, Boutet de Monvel operators on singular manifolds, C. R. Math. Acad. Sci. Paris 354 (2016), no. 3, 239–243. MR 3463018MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    _________ , Boutet de Monvel operators on Lie manifolds with boundary, Adv. Math. 312 (2017), 234–285. MR 3635812Google Scholar
  18. 18.
    Louis Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), no. 1-2, 11–51.Google Scholar
  19. 19.
    _________ , A course on pseudo differential operators and their applications, Mathematics Department, Duke University, Durham, N.C., 1976, Duke University Mathematics Series, No. II.Google Scholar
  20. 20.
    Alcides Buss, Siegfried Echterhoff, and Rufus Willett, Exotic crossed products, Operator algebras and applications—the Abel Symposium 2015, Abel Symp., vol. 12, Springer, [Cham], 2017, pp. 67–114. MR 3837592Google Scholar
  21. 21.
    _________ , Exotic crossed products and the Baum-Connes conjecture, J. Reine Angew. Math. 740 (2018), 111–159. MR 3824785Google Scholar
  22. 22.
    P. Carrillo Rouse, J. M. Lescure, and B. Monthubert, A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary, J. Topol. Anal. 6 (2014), no. 1, 27–74. MR 3190137MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Pierre Cartier, Groupoïdes de Lie et leurs algébroïdes, Astérisque (2009), no. 326, Exp. No. 987, viii, 165–196 (2010), Séminaire Bourbaki. Vol. 2007/2008. MR 2605322Google Scholar
  24. 24.
    Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890Google Scholar
  25. 25.
    Woocheol Choi and Raphael Ponge, Tangent maps and tangent groupoid for Carnot manifolds, arXiv:1510.05851, 2015.Google Scholar
  26. 26.
    _________ , Privileged coordinates and nilpotent approximation for Carnot manifolds, II. Carnot coordinates, arXiv:1703.05494, 2017.Google Scholar
  27. 27.
    _________ , Privileged coordinates and nilpotent approximation of Carnot manifolds, I. General results, arXiv:1709.09045, 2017.Google Scholar
  28. 28.
    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
  29. 29.
    _________ , An analogue of the Thom isomorphism for crossed products of aC -algebra by an action ofR, Adv. in Math. 39 (1981), no. 1, 31–55.Google Scholar
  30. 30.
    _________ , A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628.Google Scholar
  31. 31.
    _________ , 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. MR 866491 (88k:58149)Google Scholar
  32. 32.
    _________ , Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.Google Scholar
  33. 33.
    Alain Connes and Nigel Higson, Déformations, morphismes asymptotiques etK-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106. MR 1065438Google Scholar
  34. 34.
    Alain Connes and Henri Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867Google Scholar
  35. 35.
    _________ , Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, 199–246. MR 1657389Google Scholar
  36. 36.
    Alain Connes and Georges Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056Google Scholar
  38. 38.
    Claire Debord, Holonomy groupoids of singular foliations, J. Differential Geom. 58 (2001), no. 3, 467–500. MR 1906783MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Claire Debord and Jean-Marie Lescure, In preparation.Google Scholar
  40. 40.
    _________ , K-duality for pseudomanifolds with isolated singularities, J. Funct. Anal. 219 (2005), no. 1, 109–133.Google Scholar
  41. 41.
    _________ , K-duality for stratified pseudomanifolds, Geom. Topol. 13 (2009), no. 1, 49–86. MR 2469513Google Scholar
  42. 42.
    _________ , Index theory and groupoids, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2010, pp. 86–158.Google Scholar
  43. 43.
    Claire Debord, Jean-Marie Lescure, and Victor Nistor, Groupoids and an index theorem for conical pseudo-manifolds, J. Reine Angew. Math. 628 (2009), 1–35. MR 2503234Google Scholar
  44. 44.
    Claire Debord, Jean-Marie Lescure, and Frédéric Rochon, Pseudodifferential operators on manifolds with fibred corners, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1799–1880.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Claire Debord and Georges Skandalis, Adiabatic groupoid, crossed product by \(\mathbb {R}_+^\ast \)and pseudodifferential calculus, Adv. Math. 257 (2014), 66–91.MathSciNetzbMATHGoogle Scholar
  46. 46.
    _________ , Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus, arXiv:1705.09588, 2017.Google Scholar
  47. 47.
    _________ , Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps, Preprint (part of arXiv:1705.09588), 2017.Google Scholar
  48. 48.
    J. J. Duistermaat, Fourier integral operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011, Reprint of the 1996 edition [MR1362544], based on the original lecture notes published in 1973 [MR0451313]. MR 2741911Google Scholar
  49. 49.
    J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183–269. MR 0388464MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    George A. Elliott, Toshikazu Natsume, and Ryszard Nest, The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics, Comm. Math. Phys. 182 (1996), no. 3, 505–533. MR 1461941Google Scholar
  51. 51.
    Thierry Fack and Georges Skandalis, Connes’ analogue of the Thom isomorphism for the Kasparov groups, Invent. Math. 64 (1981), no. 1, 7–14.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 0578656MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Damien Gaboriau, Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1501–1527. MR 2827853Google Scholar
  54. 54.
    Gerd Grubb, Functional calculus of pseudodifferential boundary problems, second ed., Progress in Mathematics, vol. 65, Birkhäuser Boston Inc., 1996.Google Scholar
  55. 55.
    André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque (1984), no. 116, 70–97, Transversal structure of foliations (Toulouse, 1982). MR 755163Google Scholar
  56. 56.
    Nigel Higson, Vincent Lafforgue, and Georges Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354. MR 1911663 (2003g:19007)Google Scholar
  57. 57.
    Nigel Higson and John Roe, Mapping surgery to analysis. I. Analytic signatures, K-Theory 33 (2005), no. 4, 277–299. MR 2220522Google Scholar
  58. 58.
    _________ , Mapping surgery to analysis. II. Geometric signatures, K-Theory 33 (2005), no. 4, 301–324. MR 2220523Google Scholar
  59. 59.
    _________ , Mapping surgery to analysis. III. Exact sequences, K-Theory 33 (2005), no. 4, 325–346. MR 2220524Google Scholar
  60. 60.
    _________ , K-homology, assembly and rigidity theorems for relative eta invariants, Pure and Applied Mathematics Quarterly 6 (2010), no. 2, 555–601.Google Scholar
  61. 61.
    Michel Hilsum and Georges Skandalis, MorphismesK-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.MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183. MR 0383152Google Scholar
  63. 63.
    _________ , The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 0609014MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    _________ , The calculus of Fourier integral operators, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Princeton Univ. Press, Princeton, N.J., 1971, pp. 33–57. Ann. of Math. Studies, No. 70. MR 0341193Google Scholar
  65. 65.
    _________ , Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 0388463MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    _________ , The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003, Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773Google Scholar
  67. 67.
    _________ , The analysis of linear partial differential operators. II, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Differential operators with constant coefficients, Reprint of the 1983 original. MR 2108588Google Scholar
  68. 68.
    _________ , The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition. MR 2304165Google Scholar
  69. 69.
    _________ , The analysis of linear partial differential operators. IV, Classics in Mathematics, Springer-Verlag, Berlin, 2009, Fourier integral operators, Reprint of the 1994 edition. MR 2512677Google Scholar
  70. 70.
    Pierre Julg and Erik van Erp, The geometry of the osculating nilpotent group structures of the Heisenberg calculus, J. Lie Theory 28 (2018), no. 1, 107–138.MathSciNetzbMATHGoogle Scholar
  71. 71.
    Gennadi G. Kasparov, Topological invariants of elliptic operators I: K-homology, Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 39 (1975), 796–838.Google Scholar
  72. 72.
    _________ , The operatorK-functor and extensions ofC -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636.Google Scholar
  73. 73.
    Gennadi G. Kasparov and Georges Skandalis, Groups acting on buildings, operator K-theory and Novikov’s conjecture, K-theory 4 (1991), 303–337.MathSciNetCrossRefGoogle Scholar
  74. 74.
    Jean-Marie Lescure, Triplets spectraux pour les variétés à singularité conique isolé e, Bull. Soc. math. France 129 (2001), no. 4, 593–623.MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Jean-Marie Lescure, Dominique Manchon, and Stéphane Vassout, About the convolution of distributions on groupoids, J. Noncommut. Geom. 11 (2017), no. 2, 757–789. MR 3669118MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Jean-Marie Lescure and Stéphane Vassout, Fourier integral operators on Lie groupoids, Adv. Math. 320 (2017), 391–450. MR 3709110Google Scholar
  77. 77.
    Pedro T. P. Lopes and Severino T. Melo, K-theory of the Boutet de Monvel algebra with classical SG-symbols on the half space, Math. Nachr. 287 (2014), no. 16, 1804–1827. MR 3274491Google Scholar
  78. 78.
    Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005.Google Scholar
  79. 79.
    Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664. MR 1133743MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260–310.MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    _________ , Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), no. 4, 833–866, Mikio Sato: a great Japanese mathematician of the twentieth century.Google Scholar
  82. 82.
    Severino T. Melo, Thomas Schick, and Elmar Schrohe, Families index for Boutet de Monvel operators, Münster J. Math. 6 (2013), no. 2, 343–364. MR 3148215Google Scholar
  83. 83.
    Richard B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 217–234. MR 1159214Google Scholar
  84. 84.
    _________ , Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices (1992), no. 3, 51–61. MR 1154213Google Scholar
  85. 85.
    _________ , The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters Ltd., Wellesley, MA, 1993.Google Scholar
  86. 86.
    Richard B. Melrose and Paolo Piazza, AnalyticK-theory on manifolds with corners, Adv. Math. 92 (1992), no. 1, 1–26. MR 1153932Google Scholar
  87. 87.
    A. S. Mishchenko and A. T. Fomenko, The index of elliptic operators over C*-algebras, Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 43 (1979), no. 4, 831–859.Google Scholar
  88. 88.
    Omar Mohsen, On the deformation groupoid of the inhomogeneous pseudo-differential calculus, arXiv:1806.08585, 2018.Google Scholar
  89. 89.
    Bertrand Monthubert, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc. 127 (1999), no. 10, 2871–2881.MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    _________ , Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal. 199 (2003), no. 1, 243–286.MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Calvin C. Moore and Claude L. Schochet, Global analysis on foliated spaces, second ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge University Press, New York, 2006. MR 2202625Google Scholar
  93. 93.
    Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoidC -algebras, J. Operator Theory 17 (1987), no. 1, 3–22.MathSciNetzbMATHGoogle Scholar
  94. 94.
    F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275Google Scholar
  95. 95.
    Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152.MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    William L. Paschke, K-theory for commutants in the Calkin algebra, Pacific J. Math. 95 (1981), no. 2, 427–434. MR 632196MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Alan L. T. Paterson, Continuous family groupoids, Homology Homotopy Appl. 2 (2000), 89–104. MR 1782594Google Scholar
  98. 98.
    Paolo Piazza and Thomas Schick, The surgery exact sequence, K-theory and the signature operator, Ann. K-Theory 1 (2016), no. 2, 109–154. MR 3514938MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Paolo Piazza and Vito Felice Zenobi, Singular spaces, groupoids and metrics of positive scalar curvature, arXiv:1803.02697, 2018.Google Scholar
  100. 100.
    Raphaël Ponge, The tangent groupoid of a Heisenberg manifold, Pacific J. Math. 227 (2006), no. 1, 151–175.MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Jean Pradines, Feuilletages: holonomie et graphes locaux, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 13, 297–300. MR 765427Google Scholar
  102. 102.
    Jean Renault, A groupoid approach toC -algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.zbMATHCrossRefGoogle Scholar
  103. 103.
    _________ , C -algebras and dynamical systems, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2009, 27o Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]. MR 2536186Google Scholar
  104. 104.
    John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, 2003.Google Scholar
  105. 105.
    Anton Savin, Elmar Schrohe, and Sternin Boris, Elliptic operators associated with groups of quantized canonical transformations, arXiv:1612.02981.Google Scholar
  106. 106.
    Elmar Schrohe, A short introduction to Boutet de Monvel’s calculus, Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., vol. 125, Birkhäuser, Basel, 2001, pp. 85–116.zbMATHCrossRefGoogle Scholar
  107. 107.
    M. A. Shubin, Pseudodifferential operators and spectral theory, second ed., Springer-Verlag, Berlin, 2001, Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334Google Scholar
  108. 108.
    Georges Skandalis, Jean-Louis Tu, and Guoliang Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), 807–834.MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463Google Scholar
  110. 110.
    François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York-London, 1980, Pseudodifferential operators, The University Series in Mathematics. MR 597144Google Scholar
  111. 111.
    _________ , Introduction to pseudodifferential and Fourier integral operators. Vol. 2, Plenum Press, New York-London, 1980, Fourier integral operators, The University Series in Mathematics. MR 597145Google Scholar
  112. 112.
    Alain Valette, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002, From notes taken by Indira Chatterji, With an appendix by Guido Mislin. MR 1907596Google Scholar
  113. 113.
    Erik van Erp, The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I, Ann. of Math. (2) 171 (2010), no. 3, 1647–1681. MR 2680395Google Scholar
  114. 114.
    Erik van Erp and Robert Yuncken, A groupoid approach to pseudodifferential operators, J. Reine Agnew. Math. 2019 (2019), no. 756, 151–182.zbMATHGoogle Scholar
  115. 115.
    _________ , On the tangent groupoid of a filtered manifold, Bull. Lond. Math. Soc. 49 (2017), no. 6, 1000–1012.MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Stéphane Vassout, Unbounded pseudodifferential calculus on Lie groupoids, J. Funct. Anal. 236 (2006), no. 1, 161–200. MR 2227132Google Scholar
  117. 117.
    Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 0415338Google Scholar
  118. 118.
    H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75. MR 739904MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Mayuko Yamashita, A topological approach to indices of geometric operators on manifolds with fibered boundaries, arXiv:1902.03767.Google Scholar
  120. 120.
    Vito Felice Zenobi, The adiabatic groupoid and the Higson-Roe exact sequence, arXiv:1901.05081.Google Scholar

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Authors and Affiliations

  1. 1.UPMC Paris 06 CNRS, IMJ-PRG UFR de MathématiquesUniversité Paris Diderot, Sorbonne Paris Cité Sorbonne UniversitésParisFrance

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