Proceedings of the Steklov Institute of Mathematics

, Volume 271, Issue 1, pp 193–211 | Cite as

Noncommutative elliptic theory. Examples

  • A. Yu. SavinEmail author
  • B. Yu. Sternin


We study differential operators with coefficients in noncommutative algebras. As an algebra of coefficients, we consider crossed products corresponding to the action of a discrete group on a smooth manifold. We give index formulas for the Euler, signature, and Dirac operators twisted by projections over the crossed product. The index of Connes operators on the noncommutative torus is computed.


STEKLOV Institute Conjugacy Class Dirac Operator Elliptic Operator Cotangent Bundle 
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  1. 1.
    A. Connes, “C* algèbres et géométrie différentielle,” C. R. Acad. Sci. Paris, Sér. A 290(13), 599–604 (1980).MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994).zbMATHGoogle Scholar
  3. 3.
    A. Connes and G. Landi, “Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations,” Commun. Math. Phys. 221(1), 141–159 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    G. Landi and W. van Suijlekom, “Principal Fibrations from Noncommutative Spheres,” Commun. Math. Phys. 260(1), 203–225 (2005).zbMATHCrossRefGoogle Scholar
  5. 5.
    A. Connes and M. Dubois-Violette, “Noncommutative Finite-Dimensional Manifolds. I: Spherical Manifolds and Related Examples,” Commun. Math. Phys. 230(3), 539–579 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. B. Antonevich and A. V. Lebedev, “Functional Equations and Functional Operator Equations. A C*-Algebraic Approach,” Tr. S.-Peterburg.Mat. Obshch. 6, 34–140 (1998) [Am. Math. Soc. Transl., Ser. 2, 199, 25–116 (2000)].MathSciNetGoogle Scholar
  7. 7.
    A. B. Antonevich, “Boundary Value Problems with Strong Nonlocalness for Elliptic Equations,” Izv. Akad. Nauk SSSR, Ser. Mat. 53(1), 3–24 (1989) [Math. USSR, Izv. 34 (1), 1–21 (1990)].MathSciNetGoogle Scholar
  8. 8.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators (Birkhäuser, Basel, 2008), Oper. Theory: Adv. Appl. 183.zbMATHGoogle Scholar
  9. 9.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, “On the Index of Nonlocal Elliptic Operators,” Dokl. Akad. Nauk 420(5), 592–597 (2008) [Dokl. Math. 77 (3), 441–445 (2008)].Google Scholar
  10. 10.
    L. B. Schweitzer, “Spectral Invariance of Dense Subalgebras of Operator Algebras,” Int. J. Math. 4(2), 289–317 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J. J. Kohn and L. Nirenberg, “An Algebra of Pseudo-differential Operators,” Commun. Pure Appl. Math. 18, 269–305 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    P. E. Conner and E. E. Floyd, Differentiable Periodic Maps (Academic, New York, 1964).zbMATHGoogle Scholar
  13. 13.
    J. Slominska, “On the Equivariant Chern Homomorphism,” Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24, 909–913 (1976).MathSciNetzbMATHGoogle Scholar
  14. 14.
    P. Baum and A. Connes, “Chern Character for Discrete Groups,” in A Fête of Topology (Academic, Boston, MA, 1988), pp. 163–232.Google Scholar
  15. 15.
    M. F. Atiyah and I. M. Singer, “The Index of Elliptic Operators. III,” Ann. Math., Ser. 2, 87, 546–604 (1968).MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. F. Atiyah and G. B. Segal, “The Index of Elliptic Operators. II,” Ann. Math., Ser. 2, 87, 531–545 (1968).MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Ji, “Trace Invariant and Cyclic Cohomology of Twisted Group C*-Algebras,” J. Funct. Anal. 130(2), 283–292 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    S. Baaj, “Calcul pseudo-différentiel et produits croisés de C*-algébres. I,” C. R. Acad. Sci. Paris, Sér. 1, 307(11), 581–586 (1988).MathSciNetzbMATHGoogle Scholar
  19. 19.
    S. Baaj, “Calcul pseudo-différentiel et produits croisés de C*-algèbres. II,” C. R.Acad. Sci. Paris, Séer. 1, 307(12), 663–666 (1988).MathSciNetzbMATHGoogle Scholar
  20. 20.
    J. Rosenberg, “Noncommutative Variations on Laplace’s Equation,” Anal. PDE 1(1), 95–114 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M. A. Rieffel, “C*-Algebras Associated with Irrational Rotations,” Pac. J. Math. 93(2), 415–429 (1981).MathSciNetzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Leibniz Universität HannoverHannoverGermany

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