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Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

  • Anton Savin
  • Boris Sternin
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 231)

Abstract

In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader’s convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.

Keywords

Elliptic operator index index formula cyclic cohomology diffeomorphism G-operator. 

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References

  1. [1]
    A. Antonevich, M. Belousov, and A. Lebedev, Functional differential equations II C -applications, Parts 1, 2, Number 94, 95 in Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, 1998.Google Scholar
  2. [2]
    A. Antonevich and A. Lebedev, Functional-Differential Equations I C -Theory, Number 70 in Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, 1994.Google Scholar
  3. [3]
    A.B. Antonevich. Elliptic pseudodifferential operators with a finite group of shifts, Math. USSR-Izv., 7 (1973), 661–674.CrossRefGoogle Scholar
  4. [4]
    A.B. Antonevich, Linear Functional Equations, Operator Approach, Universitetskoje, Minsk, 1988.Google Scholar
  5. [5]
    A.B. Antonevich, Strongly nonlocal boundary value problems for elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat., 53(1) (1989), 3–24.MathSciNetGoogle Scholar
  6. [6]
    A.B. Antonevich and V.V. Brenner, On the symbol of a pseudodifferential operator with locally independent shifts, Dokl. Akad. Nauk BSSR, 24(10) (1980), 884–887.MathSciNetMATHGoogle Scholar
  7. [7]
    A.B. Antonevich and A.V. Lebedev, Functional equations and functional operator equations A C - algebraic approach, in Proceedings of the St. Petersburg Mathematical Society, Vol. VI, Amer. Math. Soc. Transl. Ser. 2 199, 2000, 25–116.Google Scholar
  8. [8]
    M.F. Atiyah. Elliptic Operators and Compact Groups, Lecture Notes in Mathematics 401, Springer-Verlag, Berlin, 1974.Google Scholar
  9. [9]
    M.F. Atiyah. K-Theory, Second Edition, The Advanced Book Program, Addison– Wesley, Inc., 1989.Google Scholar
  10. [10]
    M.F. Atiyah and G.B. Segal, The index of elliptic operators II, Ann. Math. 87 (1968), 531–545.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    M.F. Atiyah and I.M. Singer, The index of elliptic operators III, Ann. Math. 87 (1968), 546–604.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Ch. Babbage. An assay towards the calculus of functions, part II, Philos. Trans. of the Royal Society 106 (1816), 179–256.CrossRefGoogle Scholar
  14. [14]
    P. Baum and A. Connes, Chern character for discrete groups, in A fˆete of topology, Academic Press, Boston, MA, 1988, 163–232.Google Scholar
  15. [15]
    A.V. Bitsadze and A.A. Samarskii, On some simple generalizations of linear elliptic boundary problems, Sov. Math., Dokl. 10 (1969), 398–400.Google Scholar
  16. [16]
    B. Blackadar, K-Theory for Operator Algebras, Second Edition, Mathematical Sciences Research Institute Publications 5, Cambridge University Press, 1998.Google Scholar
  17. [17]
    T. Carleman, Sur la théorie des équations intégrales et ses applications, Verh. Internat. Math.-Kongr. Zurich. 1, 1932, 138–151.Google Scholar
  18. [18]
    P.E. Conner and E.E. Floyd, Differentiable Periodic Maps, Academic Press, New York, 1964.Google Scholar
  19. [19]
    A. Connes, C -algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290(13) (1980), A599–A604.MathSciNetGoogle Scholar
  20. [20]
    A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360.MathSciNetGoogle Scholar
  21. [21]
    A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric Methods in Operator Algebras, Pitman Res. Notes in Math. 123, Longman, Harlow, 1986.Google Scholar
  22. [22]
    A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.Google Scholar
  23. [23]
    A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds I, spherical manifolds and related examples, Comm. Math. Phys. 230(3) (2002), 539– 579.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221(1) (2001), 141–159.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    A. Connes and H. Moscovici, Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., E38, Friedr. Vieweg, Wiesbaden, 2008, 57–71.Google Scholar
  26. [26]
    I.M. Gelfand, On elliptic equations, Russian Math. Surveys 15(3) (1960), 113–127.MathSciNetCrossRefGoogle Scholar
  27. [27]
    A. Gorokhovsky, Characters of cycles, equivariant characteristic classes and Fredholm modules, Comm. Math. Phys. 208(1) (1999), 1–23.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    A. Jaffe, A. Lesniewski, and K. Osterwalder, Quantum K-theory I. The Chern character, Comm. Math. Phys. 118(1) (1988), 1–14.Google Scholar
  30. [30]
    T. Kawasaki, The index of elliptic operators over V -manifolds, Nagoya Math. J. 84 (1981), 135–157.MathSciNetMATHGoogle Scholar
  31. [31]
    Yu.A. Kordyukov, Transversally elliptic operators on G-manifolds of bounded geometry, Russian J. Math. Phys. 2(2) (1994), 175–198.MathSciNetMATHGoogle Scholar
  32. [32]
    Yu.A. Kordyukov, Transversally elliptic operators on G-manifolds of bounded geometry II, Russian J. Math. Phys. 3(1) (1995), 41–64.MathSciNetMATHGoogle Scholar
  33. [33]
    Yu.A. Kordyukov, Index theory and non-commutative geometry on foliated manifolds, Russ. Math. Surv. 64(2) (2009), 273–391.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    G. Landi and W. van Suijlekom, Principal fibrations from noncommutative spheres, Comm. Math. Phys. 260(1) (2005), 203–225.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    G. Luke, Pseudodifferential operators on Hilbert bundles, J. Diff. Equations 12 (1972), 566–589.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    A.S. Mishchenko and A.T. Fomenko, The index of elliptic operators over C -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43(4) (1979), 831–859, 967.Google Scholar
  37. [37]
    H. Moscovici, Local index formula and twisted spectral triples, in Quanta of Maths, Clay Math. Proc. 11, Amer. Math. Soc., Providence, RI, 2010, 465–500.Google Scholar
  38. [38]
    V.E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, Elliptic Theory and Noncommutative Geometry, Operator Theory: Advances and Applications 183, Birkhäuser Verlag, Basel, 2008.Google Scholar
  39. [39]
    E. Park, Index theory of Toeplitz operators associated to transformation group C - algebras, Pacific J. Math. 223(1) (2006), 159–165.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    A.L.T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
  41. [41]
    G.K. Pedersen, C -Algebras and Their Automorphism Groups, London Mathematical Society Monographs 14, Academic Press, London–New York, 1979.Google Scholar
  42. [42]
    D. Perrot. A Riemann-Roch theorem for one-dimensional complex groupoids. Comm. Math. Phys., 218(2):373–391, 2001.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    D. Perrot, Localization over complex-analytic groupoids and conformal renormalization, J. Noncommut. Geom. 3(2) (2009), 289–325.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    D. Perrot, On the Radul cocycle, Oberwolfach reports (2011), 53–55. DOI:  10.4171/OWR/2011/45.
  45. [45]
    L.E. Rossovskii, Boundary value problems for elliptic functional-differential equations with dilatation and contraction of the arguments, Trans. Moscow Math. Soc. (2001), 185–212.Google Scholar
  46. [46]
    A. Savin, E. Schrohe, and B. Sternin, On the index formula for an isometric diffeomorphism, arXiv:1112.5515, 2011.Google Scholar
  47. [47]
    A. Savin, E. Schrohe, and B. Sternin, Uniformization and an index theorem for elliptic operators associated with diffeomorphisms of a manifold, arXiv:1111.1525, 2011.Google Scholar
  48. [48]
    A. Savin and B. Sternin, Index defects in the theory of nonlocal boundary value problems and the η-invariant, Sbornik: Mathematics 195(9) (2004), 1321–1358.MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    A. Savin and B. Sternin, Index of elliptic operators for a diffeomorphism. Journal of Noncommutative Geometry, V. 7, 2013. Preliminary Version: arxiv:1106.4195, 2011.Google Scholar
  50. [50]
    A.Yu. Savin, On the index of nonlocal elliptic operators for compact Lie groups, Cent. E ur. J. Math. 9(4) (2011), 833–850.MathSciNetMATHCrossRefGoogle Scholar
  51. [51]
    A.Yu. Savin, On the index of nonlocal operators associated with a nonisometric diffeomorphism, Mathematical Notes 90(5) (2011), 701–714.MathSciNetCrossRefGoogle Scholar
  52. [52]
    A.Yu. Savin, On the symbol of nonlocal operators in Sobolev spaces, Differential Equations, 47(6) (2011), 897–900.MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    A.Yu. Savin and B.Yu. Sternin, Index of nonlocal elliptic operators over C -algebras, Dokl. Math. 79(3) (2009), 369–372.Google Scholar
  54. [54]
    A.Yu. Savin and B.Yu. Sternin, Noncommutative elliptic theory. Examples, Proceedings of the Steklov Institute of Mathematics 271 (2010), 193–211.Google Scholar
  55. [55]
    A.Yu. Savin and B.Yu. Sternin, Nonlocal elliptic operators for compact Lie groups, Dokl. Math. 81(2) (2010), 258–261.Google Scholar
  56. [56]
    A.Yu. Savin. On the index of elliptic operators associated with a diffeomorphism of a manifold, Doklady Mathematics 82(3) (2010), 884–886.Google Scholar
  57. [57]
    A.Yu. Savin and B.Yu. Sternin. On the index of noncommutative elliptic operators over C -algebras, Sbornik: Mathematics 201(3) (2010), 377–417.Google Scholar
  58. [58]
    A.Yu. Savin and B.Yu. Sternin, Nonlocal elliptic operators for the group of dilations, Sbornik: Mathematics 202(10) (2011), 1505–1536.Google Scholar
  59. [59]
    A.Yu. Savin, B.Yu. Sternin, and E. Schrohe. Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization, Dokl. Math. 84(3) (2011), 846–849.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    L.B. Schweitzer, Spectral invariance of dense subalgebras of operator algebras, Internat. J. Math. 4(2) (1993), 289–317.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    I.M. Singer, Recent applications of index theory for elliptic operators, in Partial Differential Equations, Proc. Sympos. Pure Math., Vol. XXIII, Amer. Math. Soc., Providence, R.I., 1973, 11–31.Google Scholar
  62. [62]
    A.L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin, 1997.Google Scholar
  63. [63]
    A.L. Skubachevskii, Nonclassical boundary-value problems I, Journal of Mathematical Sciences 155(2) (2008), 199–334.MathSciNetMATHCrossRefGoogle Scholar
  64. [64]
    A.L. Skubachevskii, Nonclassical boundary-value problems II, Journal of Mathematical Sciences 166(4) (2010), 377–561.MathSciNetCrossRefGoogle Scholar
  65. [65]
    J. Slominska, On the equivariant Chern homomorphism, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys., 24 (1976), 909–913.Google Scholar
  66. [66]
    B.Yu. Sternin, On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem, Cent. E ur. J. Math. 9(4) (2011), 814–832.Google Scholar
  67. [67]
    M. Vergne, Equivariant index formulas for orbifolds, Duke Math. J. 82(3) (1996), 637–652.MathSciNetMATHCrossRefGoogle Scholar
  68. [68]
    D.P. Williams, Crossed Products of C -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, Providence, RI, 2007.Google Scholar
  69. [69]
    G. Zeller-Meier, Produits croisés d’une C -algèbre par un groupe d’automorphismes, J. Math. Pures Appl. 47(9) (1968), 101–239.MathSciNetMATHGoogle Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Institut fur AnalysisLeibniz University of HannoverHannoverGermany

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