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Noncommutative elliptic theory. Examples

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Abstract

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.

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References

  1. A. Connes, “C* algèbres et géométrie différentielle,” C. R. Acad. Sci. Paris, Sér. A 290(13), 599–604 (1980).

    MathSciNet  MATH  Google Scholar 

  2. A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994).

    MATH  Google Scholar 

  3. A. Connes and G. Landi, “Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations,” Commun. Math. Phys. 221(1), 141–159 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Landi and W. van Suijlekom, “Principal Fibrations from Noncommutative Spheres,” Commun. Math. Phys. 260(1), 203–225 (2005).

    Article  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    MathSciNet  Google Scholar 

  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)].

    MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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. L. B. Schweitzer, “Spectral Invariance of Dense Subalgebras of Operator Algebras,” Int. J. Math. 4(2), 289–317 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. J. Kohn and L. Nirenberg, “An Algebra of Pseudo-differential Operators,” Commun. Pure Appl. Math. 18, 269–305 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. E. Conner and E. E. Floyd, Differentiable Periodic Maps (Academic, New York, 1964).

    MATH  Google Scholar 

  13. J. Slominska, “On the Equivariant Chern Homomorphism,” Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24, 909–913 (1976).

    MathSciNet  MATH  Google Scholar 

  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. M. F. Atiyah and I. M. Singer, “The Index of Elliptic Operators. III,” Ann. Math., Ser. 2, 87, 546–604 (1968).

    Article  MathSciNet  Google Scholar 

  16. M. F. Atiyah and G. B. Segal, “The Index of Elliptic Operators. II,” Ann. Math., Ser. 2, 87, 531–545 (1968).

    Article  MathSciNet  Google Scholar 

  17. R. Ji, “Trace Invariant and Cyclic Cohomology of Twisted Group C*-Algebras,” J. Funct. Anal. 130(2), 283–292 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  20. J. Rosenberg, “Noncommutative Variations on Laplace’s Equation,” Anal. PDE 1(1), 95–114 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  21. M. A. Rieffel, “C*-Algebras Associated with Irrational Rotations,” Pac. J. Math. 93(2), 415–429 (1981).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia, ul. Muklukho-Maklaya 6, Moscow, 117198, Russia

    A. Yu. Savin & B. Yu. Sternin

  2. Leibniz Universität Hannover, Welfengarten 1, D-30167, Hannover, Germany

    A. Yu. Savin & B. Yu. Sternin

Authors
  1. A. Yu. Savin
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  2. B. Yu. Sternin
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Correspondence to A. Yu. Savin.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 271, pp. 204–223.

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Savin, A.Y., Sternin, B.Y. Noncommutative elliptic theory. Examples. Proc. Steklov Inst. Math. 271, 193–211 (2010). https://doi.org/10.1134/S0081543810040152

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