Russian Journal of Mathematical Physics

, Volume 20, Issue 3, pp 345–359 | Cite as

Uniformization of nonlocal elliptic operators and KK-theory

  • A. Yu. Savin
  • B. Yu. Sternin


By a pseudodifferential uniformization of a nonlocal elliptic operator we mean the procedure of reducing the operator to a pseudodifferential operator with a controlled modification of the index. In the paper, we suggest an approach to solving the uniformization problem; this approach uses the reduction of the symbol of a nonlocal operator to the symbol of a pseudodifferential operator. The technical apparatus here is Kasparov’s KK-theory.


Compact Operator Elliptic Operator Short Exact Sequence Cotangent Bundle Fredholm Property 
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  1. 1.
    A. Connes, “C* algèbres et géométrie différentielle,” C. R. Acad. Sci. Paris Sér. A-B 290(13), A599–A604 (1980).MathSciNetGoogle Scholar
  2. 2.
    A. B. Antonevich and A. V. Lebedev, “Functional Equations and Functional Operator Equations. A C*-Algebraic Approach,” Proc. of the St. Petersburg Math. Soc. VI, 199 of Amer. Math. Soc. Transl. Ser. 2, 25–116 (Amer. Math. Soc., Providence, RI, 2000).Google Scholar
  3. 3.
    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).zbMATHGoogle Scholar
  4. 4.
    A. Connes and H. Moscovici, “Type III and Spectral Triples,” Traces in number theory, geometry and quantum fields, Aspects Math., E38, 57–71 (Friedr. Vieweg, Wiesbaden, 2008).MathSciNetGoogle Scholar
  5. 5.
    A. Savin and B. Sternin, “Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds,” Pseudo-Differential Operators, Generalized Functions and Asymptotics 231, OperatorTheory: Advances and Applications (Birkhäuser, 1–26, 2013) arXiv:1207.3017.Google Scholar
  6. 6.
    M. F. Atiyah and I. M. Singer, “The Index of Elliptic Operators I,” Ann. of Math. 87, 484–530 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Yu. Savin and B. Yu. Sternin, “Nonlocal Elliptic Operators for Compact Lie Groups,” Dokl. Math. 81(2), 258–261 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Yu. Sternin, “On a Class of Nonlocal Elliptic Operators for Compact Lie Groups. Uniformization and Finiteness Theorem,” Cent. Eur. J. Math. 9(4), 814–832 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Savin and B. Sternin, “Index of Elliptic Operators for Diffeomorphisms of Manifolds,” J. Noncommutative Geometry 7 (2013); arXiv:1106.4195.Google Scholar
  10. 10.
    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), 846–849 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A.S. Mishchenko, “Infinite-Dimensional Representations of Discrete Groups, and Higher Signatures,” Mathematics of the USSR-Izvestiya 8(1), 85–111 (1974).ADSCrossRefGoogle Scholar
  12. 12.
    G. Kasparov, “Equivariant KK-Theory and the Novikov Conjecture,” Inv. Math. 91(1), 147–201 (1988).MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Yu. Savin and B.Yu. Sternin, “Uniformization of Nonlocal Elliptic Operators and KK-Theory,” Dokl. Math. 87(1), 20–22 (2013).CrossRefzbMATHGoogle Scholar
  14. 14.
    D. P. Williams, Crossed Products of C*-Algebras (Mathematical Surveys and Monographs 134, American Mathematical Society, Providence, RI, 2007).CrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New-York London, 1962).zbMATHGoogle Scholar
  17. 17.
    G. Luke, “Pseudodifferential Operators on Hilbert Bundles,” J. Differential Equations 12, 566–589 (1972).MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    B. Blackadar, K-Theory for Operator Algebras (Number 5 in Mathematical Sciences Research Institute Publications. Cambridge University Press, 1998, Second edition).zbMATHGoogle Scholar
  19. 19.
    A. Connes, G. Skandalis, “The Longitudinal Index Theorem for Foliations,” Publ. Res. Inst. Math. Sci. 20(6), 1139–1183 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. A. Wolf, “Essential Self-Adjointness for the Dirac Operator and Its Square,” Indiana Univ. Math. J. 22, 611–640 (1972/73).MathSciNetCrossRefGoogle Scholar
  21. 21.
    N. Higson, G. Kasparov,“E-Theory and KK-Theory for Groups which Act Properly and Isometrically on Hilbert Space,” Invent. Math. 144(1), 23–74 (2001).MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    J.-L. Tu, “The Gamma Element for Groups Which Admit a Uniform Embedding into Hilbert Space,” Operator Theory: Advances and Applications 153 (Birkhäuser Basel,, 2004), pp. 271–286.CrossRefGoogle Scholar
  23. 23.
    G. G. Kasparov, “The Operator K-Functor and Extentions of C*-Algebras,” Math. USSR, Izv. 16, 513–672 (1981).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • A. Yu. Savin
    • 1
  • B. Yu. Sternin
    • 1
  1. 1.Peoples’ Friendship University of RussiaMoscowRussia

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