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Journal of Mathematical Sciences

, Volume 164, Issue 4, pp 603–636 | Cite as

Noncommutative geometry and classification of elliptic operators

  • V. E. Nazaikinskii
  • A. Yu. Savin
  • B. Yu. Sternin
Article

Abstract

The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated in terms of K-theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it is based on the concept of abstract elliptic operators and is given in term of K-homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with conic singularities, ribs, and general so-called stratified manifolds: it suffices just to replace the phrase “smooth manifold” by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the symbol components corresponding to strata of positive codimensions are operator-valued functions), but the solution of the classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a K-group of a noncommutative C* -algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial. Note that the proofs are based on noncommutative geometry (more exactly, the K-theory of C* -algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods of noncommutative geometry (in particular, the localization principle in C* -algebras).

Keywords

Manifold Elliptic Operator Smooth Manifold Fredholm Operator Noncommutative Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Uspekhi Mat. Nauk, 19, No. 3(117), 53–161 (1964).zbMATHGoogle Scholar
  2. 2.
    A. Antonevich and A. Lebedev, Functional Differential Equations. I. C*-Theory, Longman, Harlow (1994).Google Scholar
  3. 3.
    A. Antonevich and A. Lebedev, Functional Differential Equations. II. C*-Applications. Parts 1-2, Longman, Harlow (1998).Google Scholar
  4. 4.
    W. Arveson, An Invitation to C*-Algebras, Springer-Verlag, New York–Heidelberg–Berlin (1976).zbMATHGoogle Scholar
  5. 5.
    M. F. Atiyah, “Global theory of elliptic operators,” Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 21–30 (1970).Google Scholar
  6. 6.
    M. F. Atiyah and R. Bott, “The index problem for manifolds with boundary,” Differential Analysis, Bombay Colloq., Oxford Univ. Press, London, 175–186 (1964).Google Scholar
  7. 7.
    M. F. Atiyah and I.M. Singer, “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc., 69, 422–433 (1963).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Baum and R.G. Douglas, “K-homology and index theory,” Operator algebras and applications, Part I (Kingston, Ont., 1980), Amer. Math. Soc., Providence, R.I., 117–173 (1982).Google Scholar
  9. 9.
    B. Blackadar, K-Theory for Operator Algebras, Cambridge University Press, Cambridge (1998).zbMATHGoogle Scholar
  10. 10.
    L. Brown, R. Douglas, and P. Fillmore, “Extensions of C*-algebras and K-homology,” Ann. Math. 2, 105, 265–324 (1977).CrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Dauns and K.H. Hofmann, Representation of Rings by Sections, AMS, Providence (1968).Google Scholar
  12. 12.
    J. Dixmier, Les C*-Algebres et Leurs Representations, Gauthier-Villars, Paris (1969).Google Scholar
  13. 13.
    Yu. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Birkhäuser, Boston–Basel–Berlin (1997).zbMATHGoogle Scholar
  14. 14.
    I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. I, II, Birkhäuser, Basel (1992).Google Scholar
  15. 15.
    P. Haskell, “Index theory of geometric Fredholm operators on varieties with isolated singularities,” K-Theory, 1, No. 5, 457–466 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    N. Higson and J. Roe, Analytic K-Homology, Oxford University Press, Oxford (2000).zbMATHGoogle Scholar
  17. 17.
    G. G. Kasparov, “Topological invariants of elliptic operators. I,” Izv. Math., 9, No. 4, 751–792 (1976).zbMATHMathSciNetGoogle Scholar
  18. 18.
    G. Kasparov, “Equivariant KK-theory and the Novikov conjecture,” Inv. Math., 91, No. 1, 147–201 (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. A. Kondratiev, “Boundary-value problems for elliptic equations in domains with conical and angular points,” Tr. Mosk. Mat. Obs., 16, 209–292 (1967).Google Scholar
  20. 20.
    R. Lauter and S. Moroianu, “The index of cusp operators on manifolds with corners,” Ann. Global Anal. Geom., 21, No. 1, 31–49 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P.-Y. Le Gall and B. Monthubert, “K-theory of the indicial algebra of a manifold with corners,” K-Theory, 23, No. 2, 105–113 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    V.P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).Google Scholar
  23. 23.
    R. Mazzeo, “Elliptic theory of differential edge operators. I,” Comm. Partial Differential Equations, 10, 1615–1664 (1991).MathSciNetGoogle Scholar
  24. 24.
    S. T. Melo, R. Nest, and E. Schrohe, “C*-structure and K-theory of Boutet de Monvel’s algebra,” J. Reine Angew. Math., 561, 145–175 (2003).zbMATHMathSciNetGoogle Scholar
  25. 25.
    R. Melrose, “Transformation of boundary problems,” Acta Math., 147, 149–236 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. Melrose, “Analysis on manifolds with corners. Lecture Notes,” Preprint MIT, Cambrige (1988).Google Scholar
  27. 27.
    R. Melrose, “Pseudodifferential operators, corners, and singular limits,” Proc. Internat. Cong. of Mathematicians (Kyoto), Springer, Berlin–Heidelberg–New York, 217–234 (1990).Google Scholar
  28. 28.
    R. Melrose and G.A. Mendoza, “Elliptic boundary problems in spaces with conical points,” Proc. Journees “Equ. D´eriv. Part.”, St. Jean-de-Monts, Conf. 4, 1–21 (1981).Google Scholar
  29. 29.
    R. Melrose and V. Nistor, “K-theory of C*-algebras of b-pseudodifferential operators,” Geom. Funct. Anal., 8, No. 1, 88–122 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    R. Melrose and P. Piazza, “Analytic K-theory on manifolds with corners,” Adv. Math., 92, No. 1, 1–26 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    R. Melrose and F. Rochon, “Index in K-theory for families of fibred cusp operators,” ArXiv: math.DG/0507590 (2005).Google Scholar
  32. 32.
    B. Monthubert, “Pseudodifferential calculus on manifolds with corners and groupoids,” Proc. Amer. Math. Soc., 127, No. 10, 2871–2881 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    B. Monthubert, “Groupoids of manifolds with corners and index theory,” Contemp. Math., 282, 147–157 (2001).MathSciNetGoogle Scholar
  34. 34.
    B. Monthubert, “Groupoids and pseudodifferential calculus on manifolds with corners,” J. Funct. Anal., 199, No. 1, 243–286 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    B. Monthubert and V. Nistor, “A topological index theorem for manifolds with corners,” ArXiv: math.KT/0507601 (2005).Google Scholar
  36. 36.
    V. Nazaikinskii, G. Rozenblioum, A. Savin, and B. Sternin, “Guillemin transform and Toeplitz representations for operators on singular manifolds,” Spectral Geometry of Manifolds with Boundary, AMS, Providence, 281–306 (2005).Google Scholar
  37. 37.
    V. Nazaikinskii, A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Theory on Singular Manifolds, CRC-Press, Boca Raton (2005).Google Scholar
  38. 38.
    V. Nazaikinskii, A. Savin, and B. Sternin, “Elliptic theory on manifolds with corners. I. Dual manifolds and pseudodifferential operators,” ArXiv: math.OA/0608353 (2006).Google Scholar
  39. 39.
    V. Nazaikinskii, A. Savin, and B. Sternin, “Elliptic theory on manifolds with corners. II. Homotopy classification of elliptic operators,” ArXiv: math.OA/0608354 (2006).Google Scholar
  40. 40.
    V. Nazaikinskii, A. Savin, and B. Sternin, “On the homotopy classification of elliptic operators on stratified manifolds,” ArXiv: math.KT/0608332 (2006).Google Scholar
  41. 41.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “On the homotopy classification of elliptic operators on stratified manifolds,” Dokl. Akad. Nauk, 408, No. 5, 591–595 (2006).MathSciNetGoogle Scholar
  42. 42.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “Pseudodifferential operators on stratified manifolds. I, II,” Differ. Equ., 45, No. 4-5, 536–549, 685–696 (2007).CrossRefMathSciNetGoogle Scholar
  43. 43.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “On the homotopy classification of elliptic operators on manifolds with corners,” Dokl. Math., 75, No. 2, 186–189 (2007).zbMATHCrossRefGoogle Scholar
  44. 44.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “On general localization principle in C*- algebras,” to appear.Google Scholar
  45. 45.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “Homotopy classification of elliptic operators on manifolds with corners,” to appear.Google Scholar
  46. 46.
    V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin, and B.-V. Shul’tse, “On the index of elliptic operators on manifolds with edges,” Sb. Math., 196, No. 9-10, 1271–1305 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    W. L. Paschke, “K-theory for commutants in the Calkin algebra,” Pacific J. Math., 95, No. 2, 427–434 (1981).zbMATHMathSciNetGoogle Scholar
  48. 48.
    G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London–New York (1979).zbMATHGoogle Scholar
  49. 49.
    B. A. Plamenevskiĭ and V. N. Senichkin, “Solvable operator algebras,” St. Petersburg Math. J., 6, No. 5, 895–968 (1995).MathSciNetGoogle Scholar
  50. 50.
    B. A. Plamenevskiĭ and V. N. Senichkin, “On a class of pseudodifferential operators on ℝm and on stratified manifolds,” Sb. Math., 191, No. 5-6, 725–757 (2000).CrossRefMathSciNetGoogle Scholar
  51. 51.
    B. A. Plamenevskiĭ and V. N. Senichkin, “Representations of C*-algebras of pseudodifferential operators on piecewise-smooth manifolds,” St. Petersburg Math. J., 13, No. 6, 993–1032 (2002).MathSciNetGoogle Scholar
  52. 52.
    A. Savin, “Elliptic operators on singular manifolds and K-homology,” K-Theory, 34, No. 1, 71–98 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    B.-W. Schulze, Pseudodifferential Operators on Manifolds with Singularities, North-Holland, Amsterdam (1991).Google Scholar
  54. 54.
    B.-W. Schulze, B. Sternin, and V. Shatalov, “Structure rings of singularities and differential equations,” Differential Equations, Asymptotic Analysis, and Mathematical Physics, Akademie Verlag, Berlin, 325–347 (1996).Google Scholar
  55. 55.
    B.-W. Schulze, B. Sternin, and V. Shatalov, Differential Equations on Singular Manifolds. Semiclassical Theory and Operator Algebras, Wiley-VCH Verlag, Berlin–New York (1998).Google Scholar
  56. 56.
    I. B. Simonenko, “A new general method of investigating linear operator equations of singular integral equation type. I, II,” Izv. Akad. Nauk SSSR Ser. Mat., 29, 567–586, 757–782 (1965).MathSciNetGoogle Scholar
  57. 57.
    N. L. Vasilevskiĭ, “Locality principles in operator theory,” Abstr. Northern-Caucasus Regional Conference “Linear Operators in Function Spaces,” Grozny (1989), 32-33.Google Scholar
  58. 58.
    N. Vasilevski, “Convolution operators on standard CR-manifolds. II. “Algebras of convolution operators on the Heisenberg group,” Integr. Equat. Oper. Theory, 19, No. 3, 327–348 (1994).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • V. E. Nazaikinskii
    • 1
  • A. Yu. Savin
    • 2
  • B. Yu. Sternin
    • 2
  1. 1.Institute of Mechanics Problems RANUfaRussia
  2. 2.Independent Moscow University, Russian State Social UniversityMoscowRussia

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