Results in Mathematics

, 74:160 | Cite as

The Fredholm Property for Groupoids is a Local Property

  • Rémi CômeEmail author


Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that “the Fredholm property is local”. Let \({\mathcal {G}}\rightrightarrows X\) be a topological groupoid and \((U_i)_{i\in I}\) be an open cover of X. We show that \({\mathcal {G}}\) is a Fredholm groupoid if, and only if, its reductions \({\mathcal {G}}^{U_i}_{U_i}\) are Fredholm groupoids for all \(i \in I\). We exploit this criterion to show that many groupoids encountered in practical applications are Fredholm. As an important intermediate result, we use an induction argument to show that the primitive spectrum of \(C^*({\mathcal {G}})\) can be written as the union of the primitive spectra of all \(C^*({\mathcal {G}}^{U_i}_{U_i})\), for \(i \in I\).


Fredholm operator Fredholm groupoid \(C^*\)-algebra pseudodifferential operator primitive spectrum 

Mathematics Subject Classification

58J40 (primary) 58H05 46L05 47L80 



The author would like to thank Victor Nistor for useful discussions and suggestions.


  1. 1.
    Ammann, B., Lauter, R., Nistor, V.: On the geometry of Riemannian manifolds with a Lie structure at infinity. Int. J. Math. Math. Sci. 1–4, 161–193 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Philos. Soc. 79(1), 71–99 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I. Ann. Math. Second Ser. 87, 484–530 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bigonnet, B., Pradines, J.: Graphe d’un feuilletage singulier. C. R. Acad. Sci. Paris Sér. I Math. 300(13), 439–442 (1985)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bohlen, K., Schrohe, E.: Getzler rescaling via adiabatic deformation and a renormalized index formula. J. Math. Pures Appl. 9(120), 220–252 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bohlen, K., Schulz, R.: Quantization on manifolds with an embedded submanifold (2017). arXiv e-prints, arXiv:1710.02294
  7. 7.
    Carvalho, C., Côme, R., Qiao, Y.: Gluing action groupoids: Fredholm conditions and layer potentials (2018). arXiv e-prints, arXiv:1811.07699v1
  8. 8.
    Carvalho, C., Nistor, V., Qiao, Y.: Fredholm conditions on non-compact manifolds: theory and examples. In: André, C., Bastos, M., Karlovich, A., Silbermann, B., Zaballa, I. (eds.) Operator Theory, Operator Algebras, and Matrix Theory, Operator Theory: Advances and Applications, vol. 267 , pp. 79–122. Birkhäuser, Cham (2018) CrossRefGoogle Scholar
  9. 9.
    Carvalho, C., Qiao, Y.: Layer potentials \(C^*\)-algebras of domains with conical points. Cent. Eur. J. Math. 11(1), 27–54 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Connes, A.: A survey of foliations and operator algebras. In: Operator Algebras and Applications, Part I (Kingston, Ont., 1980), Proceedings of Symposia in Pure Mathematics, vol. 38, pp. 521–628. American Mathematical Society, Providence (1982)Google Scholar
  11. 11.
    Crainic, M.: Cyclic cohomology of étale groupoids: the general case. \(K\)-Theory 17(4), 319–362 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. (2) 157(2), 575–620 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dauge, M.: Elliptic boundary value problems on corner domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988). (Smoothness and asymptotics of solutions) Google Scholar
  14. 14.
    Debord, C.: Holonomy groupoids of singular foliations. J. Differ. Geom. 58(3), 467–500 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Debord, C., Lescure, J.-M., Rochon, F.: Pseudodifferential operators on manifolds with fibred corners. Université de Grenoble. Annales de l’Institut Fourier 65(4), 1799–1880 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Debord, C., Skandalis, G.: Adiabatic groupoid, crossed product by \(\mathbb{R}_{+}^{\ast }\) and pseudodifferential calculus. Adv. Math. 257, 66–91 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Debord, C., Skandalis, G.: Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus (2017). arXiv e-prints, arXiv:1705.09588v2
  18. 18.
    Dixmier, J.: \(C^*\)-algebras. North-Holland Publishing Co., Amsterdam (1977). Translated from the French by Francis Jellett, North-Holland Mathematical Library, vol. 15 (1977)Google Scholar
  19. 19.
    Georgescu, V., Iftimovici, A.: Localizations at infinity and essential spectrum of quantum Hamiltonians. I. General theory. Rev. Math. Phys. 18(4), 417–483 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Mathematics Lecture Series, vol. 11. Publish or Perish, Inc., Wilmington (1984)Google Scholar
  21. 21.
    Gualtieri, M., Li, S.: Symplectic groupoids of log symplectic manifolds. Int. Math. Res. Not. IMRN 11, 3022–3074 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hebey, E., Robert, F.: Sobolev spaces on manifolds. In: Handbook of Global Analysis, pp. 375–415. Elsevier Sci. B. V., Amsterdam (2008)CrossRefGoogle Scholar
  23. 23.
    Hilsum, M., Skandalis, G.: 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(3), 325–390 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hörmander, L.: The analysis of linear partial differential operators. III. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274. Springer, Berlin (1985). (Pseudodifferential operators) (1985)Google Scholar
  25. 25.
    Ionescu, M., Williams, D.: The generalized Effros–Hahn conjecture for groupoids. Indiana Univ. Math. J. 58(6), 2489–2508 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Joyce, D.: On manifolds with corners. In: Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM), vol. 21, pp. 225–258. Int. Press, Somerville (2012)Google Scholar
  27. 27.
    Khoshkam, M., Skandalis, G.: Regular representation of groupoid \(C^*\)-algebras and applications to inverse semigroups. J. Reine Angew. Math. 546, 47–72 (2002)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kondrat’ ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč 16, 209–292 (1967)MathSciNetGoogle Scholar
  29. 29.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  30. 30.
    Lauter, R.: Pseudodifferential analysis on conformally compact spaces. Mem. Am. Math. Soci. 163(777), xvi+92 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lauter, R., Monthubert, B., Nistor, V.: Pseudodifferential analysis on continuous family groupoids. Doc. Math. 5, 625–655 (2000)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series, vol. 124. Cambridge University Press, Cambridge (1987)Google Scholar
  33. 33.
    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Melrose, R.B.: The Atiyah–Patodi–Singer index theorem. Research Notes in Mathematics, vol. 4 . A K Peters, Ltd., Wellesley (1993)zbMATHCrossRefGoogle Scholar
  35. 35.
    Melrose, R.B.: Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge (1995)Google Scholar
  36. 36.
    Mougel, J., Prudhon, N.: Exhaustive families of representations of \(C^\ast \)-algebras associated with \(N\)-body Hamiltonians with asymptotically homogeneous interactions. C. R. Math. Acad. Sci. Paris 357(2), 200–204 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Măntoiu, M.: C*-algebraic spectral sets, twisted groupoids and operators (2018). arXiv e-prints, arXiv:1809.03347v2
  38. 38.
    Muhly, P.S., Renault, J.N., Williams, D.P.: Equivalence and isomorphism for groupoid \(C^\ast \)-algebras. J. Oper. Theory 17(1), 3–22 (1987)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Muhly, P.S., Renault, J.N., Williams, D.P.: Continuous-trace groupoid \(C^\ast \)-algebras. III. Trans. Am. Math. Soc. 348(9), 3621–3641 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Măntoiu, M.: \(C^\ast \)-algebras, dynamical systems at infinity and the essential spectrum of generalized Schrödinger operators. J. Reine Angew. Math. 550, 211–229 (2002)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Măntoiu, M., Purice, R., Richard, S.: Spectral and propagation results for magnetic Schrödinger operators; a \(C^*\)-algebraic framework. J. Funct. Anal. 250(1), 42–67 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Nistor, V., Prudhon, N.: Exhaustive families of representations and spectra of pseudodifferential operators. J. Oper. Theory 78(2), 247–279 (2017)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on differential groupoids. Pacific J. Math. 189(1), 117–152 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Paterson, A.L.T.: Continuous family groupoids. Homol. Homotopy Appl. 2, 89–104 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Paterson, A.L.T.: The analytic index for proper, Lie groupoid actions. Groupoids in Analysis. Geometry, and Physics (Boulder, CO, 1999), Contemporary Mathematics, vol. 282, pp. 115–135. American Mathematical Society, Providence (2001)Google Scholar
  46. 46.
    Rabinovich, V., Schulze, B.-W., Tarkhanov, N.: Boundary value problems in oscillating cuspidal wedges. Rocky Mt. J. Math. 34(4), 1399–1471 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Raeburn, I., Williams, D.P.: Morita Equivalence and Continuous-trace \(C^*\)-Algebras, Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  48. 48.
    Renault, J.: A groupoid approach to \(C^{\ast } \)-algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)Google Scholar
  49. 49.
    Renault, J.: Induced representations and hypergroupoids. SIGMA Symmetry Integr. Geom. Methods Appl. 10:Paper 057,18 (2014)Google Scholar
  50. 50.
    Rieffel, M.A.: Induced representations of \(C^{\ast } \)-algebras. Adv. Math. 13, 176–257 (1974)zbMATHCrossRefGoogle Scholar
  51. 51.
    Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities, Studies in Mathematics and its Applications, vol. 24. North-Holland Publishing Co., Amsterdam (1991)Google Scholar
  52. 52.
    Tu, J.-L.: Non-Hausdorff groupoids, proper actions and \(K\)-theory. Doc. Math. 9, 565–597 (2004)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Van Erp, E., Yuncken, R.: A groupoid approach to pseudodifferential operators (2015). arXiv e-prints, arXiv:1511.01041
  54. 54.
    Van Erp, E., Yuncken, R.: On the tangent groupoid of a filtered manifold. Bull. Lond. Math. Soc. 49(6), 1000–1012 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Vasy, A.: Propagation of singularities in many-body scattering. Ann. Sci. École Norm. Sup. (4) 34(3), 313–402 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Williams, D.P.: Crossed Products of \(C^\ast \)-algebras, Mathematical Surveys and Monographs, vol. 134. American Mathematical Society, Providence (2007)Google Scholar

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Authors and Affiliations

  1. 1.Institut Élie Cartan de LorraineUniversité de LorraineMetzFrance

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