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Pseudodifferential Subspaces and Their Applications in Elliptic Theory

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Book cover C*-algebras and Elliptic Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah73-Patodi— Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.

The work was partially supported by RFBR grants NN 05-01-00982, 03-02-16336, 06-01-00098 and presidential grant MK-1713.2005.1.

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References

  1. M.S. Agranovich. Elliptic operators on closed manifolds. In Partial differential equations. VI, volume 63 of Encycl. Math. Sci., pages 1–130, 1994.

    MathSciNet  Google Scholar 

  2. M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry I. Math. Proc. Cambridge Philos. Soc., 77 (1975), 43–69.

    MathSciNet  MATH  Google Scholar 

  3. M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry II. Math. Proc. Cambridge Philos. Soc., 78 (1976), 405–432.

    MathSciNet  Google Scholar 

  4. M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry III. Math. Proc. Cambridge Philos. Soc., 79 (1976), 71–99.

    Article  MathSciNet  MATH  Google Scholar 

  5. M.F. Atiyah. The logarithm of the Dedekind η-function. Math. Annalen, 278 (1987), 335–380.

    Article  MathSciNet  MATH  Google Scholar 

  6. M.F. Atiyah. K-Theory. The Advanced Book Program. Addison-Wesley, Inc., second edition, 1989.

    Google Scholar 

  7. M.F. Atiyah and R. Bott. The index problem for manifolds with boundary. In Bombay Colloquium on Differential Analysis, pages 175–186, Oxford, 1964.

    Google Scholar 

  8. M.F. Atiyah, R. Bott, and V.K. Patodi. On the heat equation and the index theorem. Invent. Math., 19 (1973), 279–330.

    Article  MathSciNet  MATH  Google Scholar 

  9. M.F. Atiyah and I.M. Singer. The index of elliptic operators I. Ann. of Math., 87 (1968), 484–530.

    Article  MathSciNet  Google Scholar 

  10. M.F. Atiyah and I.M. Singer. Index theory for skew-adjoint Fredholm operators. Publ. Math. IHES, 37 (1969), 5–26.

    MathSciNet  MATH  Google Scholar 

  11. M.F. Atiyah and I.M. Singer. The index of elliptic operators IV. Ann. Math., 93 (1971), 119–138.

    Article  MathSciNet  Google Scholar 

  12. A. Bahri and P. Gilkey. The eta invariant, Pin c bordism, and equivariant Spin c bordism for cyclic 2-groups. Pacific Jour. Math., 128 (1987), no. 1, 1–24.

    MathSciNet  MATH  Google Scholar 

  13. M.S. Birman and M.Z. Solomyak. On the subspaces admitting a pseudodifferential projection. Vestnik LGU, 1 (1982), 18–25.

    MathSciNet  Google Scholar 

  14. M.Sh. Birman and M.Z. Solomyak. The asymptotic behavior of the spectrum of variational problems on solutions of elliptic systems. Zapiski LOMI, 115 (1982), 23–39.

    MathSciNet  MATH  Google Scholar 

  15. J.-M. Bismut. Local index theory, eta invariants and holomorphic torsion: A survey. Surveys in differential geometry, volume 3, pages 1–76, Cambridge, 1998.

    MathSciNet  Google Scholar 

  16. J.-M. Bismut and D.S. Freed. The analysis of elliptic families. I. Commun. Math. Phys., 106 (1986), 159–176.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Blackadar. K-Theory for Operator Algebras. Cambridge University Press, 1998. Second edition.

    Google Scholar 

  18. B. Booß-Bavnbek and K. Wojciechowski. Elliptic Boundary Problems for Dirac Operators. Birkhäuser, Boston-Basel-Berlin, 1993.

    MATH  Google Scholar 

  19. R. Bott and L. Tu. Differential Forms in Algebraic Topology, volume 82 of Graduate Texts in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1982.

    Google Scholar 

  20. B. Botvinnik. Manifolds with singularities accepting a metric of positive scalar curvature. Geom. Topol., 5 (2001), 683–718.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Botvinnik and P. Gilkey. The Gromov-Lawson-Rosenberg conjecture: The twisted case. Houston J. Math., 23 (1997), no. 1, 143–160.

    MathSciNet  MATH  Google Scholar 

  22. B.I. Botvinnik, V.M. Bukhshtaber, S.P. Novikov, and S.A. Yuzvinskii. Algebraic aspects of multiplication theory in complex cobordisms. Uspekhi Mat. Nauk, 55 (2000), no. 4, 5–24.

    MathSciNet  Google Scholar 

  23. L. Boutet de Monvel and V. Guillemin. The spectral theory of Toeplitz operators, volume 99 of Ann. of Math. Studies. Princeton University Press, Princeton, 1981.

    Google Scholar 

  24. L. Brown, R. Douglas, and P. Fillmore. Unitary equivalence modulo the compact operators and extensions of C*-algebras, volume 345 of Lecture Notes in Math. 1973.

    Google Scholar 

  25. L. Brown, R. Douglas, and P. Fillmore. Extensions of C*-algebras and K-homology. Ann. Math. II, 105 (1977), 265–324.

    Article  MathSciNet  Google Scholar 

  26. A.P. Calderón. Boundary value problems for elliptic equations. Outlines of the Joint Soviet-American Symposium on Partial Differential Equations, 303–304, 1963.

    Google Scholar 

  27. C. Carvalho. A K-theory proof of the cobordism invariance of the index. available at math.KT/0408260, to appear in K-theory.

    Google Scholar 

  28. A.A. Dezin. Multidimensional Analysis and Discrete Models. CRC-Press, Boca Raton, Florida, USA, 1995. English transl.: Nauka, Moscow, 1990.

    MATH  Google Scholar 

  29. J. Eells Jr. and N.H. Kuiper. An invariant for certain smooth manifolds. Ann. Mat. Pura Appl. (4), 60 (1962), 93–110.

    MathSciNet  Google Scholar 

  30. D. Freed. ℤ/k manifolds and families of Dirac operators. Invent. Math., 92 (1988), no. 2, 243–254.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Freed and R. Melrose. A mod k index theorem. Invent. Math., 107 (1992), no. 2, 283–299.

    Article  MathSciNet  MATH  Google Scholar 

  32. P.B. Gilkey. The residue of the global eta function at the origin. Adv. in Math., 40 (1981), 290–307.

    Article  MathSciNet  MATH  Google Scholar 

  33. P.B. Gilkey. The eta invariant for even-dimensional Pin c manifolds. Adv. in Math., 58 (1985), 243–284.

    Article  MathSciNet  MATH  Google Scholar 

  34. P.B. Gilkey. The eta invariant and non-singular bilinear products on R n. Can. Math. Bull., 30 (1987), 147–154.

    MathSciNet  MATH  Google Scholar 

  35. P.B. Gilkey. The eta invariant of even order operators. Lecture Notes in Mathematics, 1410:202–211, 1989.

    Article  MathSciNet  Google Scholar 

  36. P.B. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer index theorem. CRC Press, Boca Raton, FL, second edition, 1995.

    MATH  Google Scholar 

  37. I.Ts. Gohberg and M.G. Krein. Systems of integral equations on the half-line with kernels depending on the difference of the arguments. Uspekhi Matem. Nauk, 13 (1958), no. 2, 3–72.

    MathSciNet  Google Scholar 

  38. G. Grubb. A resolvent approach to traces and zeta Laurent expansions. Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., 366, pages 67–93. Amer. Math. Soc., Providence, RI, 2005.

    Google Scholar 

  39. G. Grubb and E. Schrohe. Traces and quasi-traces on the Boutet de Monvel algebra. Ann. Inst. Fourier (Grenoble), 54 (2004), no. 5, 1641–1696.

    MathSciNet  MATH  Google Scholar 

  40. G. Grubb and R.T. Seeley. Zeta and eta functions for Atiyah-Patodi-Singer operators. J. Geom. Anal., 6 (1996), no. 1, 31–77.

    MathSciNet  MATH  Google Scholar 

  41. L. Hörmander. The Analysis of Linear Partial Differential Operators. III. Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985.

    MATH  Google Scholar 

  42. G.G. Kasparov. The generalized index of elliptic operators. Funct. Anal. Appl., 7 (1973), 238–240.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Kohn and L. Nirenberg. An algebra of pseudo-differential operators. Comm. Pure Appl. Math., 18 (1965), 269–305.

    MathSciNet  MATH  Google Scholar 

  44. M. Kontsevich and S. Vishik. Geometry of determinants of elliptic operators. In Functional analysis on the eve of the 21’st century, volume 131 of Progr. Math., pages 173–197. Birkhäuser, Boston, 1995.

    Google Scholar 

  45. M. Kreck and S. Stolz. Nonconnected moduli spaces of positive sectional curvature metrics. J. Amer. Math. Soc., 6 (1993), no. 4, 825–850.

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Lauter and S. Moroianu. An index formula on manifolds with fibered cusp ends. J. of Geom. Anal., 15 (2005), no. 2, 261–283.

    MathSciNet  MATH  Google Scholar 

  47. R. Melrose. The Atiyah—Patodi—Singer Index Theorem. Research Notes in Mathematics. A.K. Peters, Boston, 1993.

    Google Scholar 

  48. R. Melrose. The eta invariant and families of pseudodifferential operators. Math. Research Letters, 2 (1995), no. 5, 541–561.

    MathSciNet  MATH  Google Scholar 

  49. O.K. Mironov. Existence of multiplicative structures in the theory of cobordism with singularities. Izv. Akad. Nauk SSR Ser. Mat., 39 (1975), no. 5, 1065–1092.

    MathSciNet  Google Scholar 

  50. J. Morgan and D. Sullivan. The transversality characteristic class and linking cycles in surgery theory. Ann. of Math., II. Ser, 99 (1974), 463–544.

    Article  MathSciNet  Google Scholar 

  51. S. Moroianu. Homology and residues of adiabatic pseudodifferential operators. Nagoya Math. J., 175 (2004), 171–221.

    MathSciNet  MATH  Google Scholar 

  52. W. Müller. Eta-invariant (some recent developments). Sem. Bourbaki. Astérisque, 227 (1994), 335–364.

    Google Scholar 

  53. V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov. Spectral boundary value problems and elliptic equations on singular manifolds. Differents. Uravnenija, 34 (1998), no. 5, 695–708. English trans.: Differential Equations, 34 (1998), no. 5, 696–710.

    MathSciNet  Google Scholar 

  54. V. Nistor. Asymptotics and index for families invariant with respect to a bundle of Lie groups. Rev. Roum. Math. Pures Appl., 47 (2002), no. 4, 451–483.

    MathSciNet  MATH  Google Scholar 

  55. S.P. Novikov. Pontrjagin classes, the fundamental group and some problems of stable algebra. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 147–155. Springer, New York, 1970.

    Google Scholar 

  56. R.S. Palais. Seminar on the Atiyah-Singer index theorem. Princeton Univ. Press, Princeton, NJ, 1965.

    MATH  Google Scholar 

  57. J. Phillips. Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull., 39 (1996), no. 4, 460–467.

    MathSciNet  MATH  Google Scholar 

  58. S. Rosenberg. Nonlocal invariants in index theory. Bull. AMS, 34 (1997), no. 4, 423–433.

    Article  MATH  Google Scholar 

  59. A. Savin, B.-W. Schulze, and B. Sternin. On invariant index formulas for spectral boundary value problems. Differentsial’nye uravnenija, 35 (1999), no. 5, 709–718.

    MathSciNet  MATH  Google Scholar 

  60. A. Savin, B.-W. Schulze, and B. Sternin. Elliptic Operators in Subspaces and the Eta Invariant. K-theory, 27 (2002), no. 3, 253–272.

    Article  MathSciNet  MATH  Google Scholar 

  61. A. Savin and B. Sternin. To the problem of homotopy classification of the elliptic boundary value problems. Doklady Mathematics, 63 (2001), no. 2, 174–178.

    MathSciNet  Google Scholar 

  62. A. Savin and B. Sternin. The eta-invariant and Pontryagin duality in K-theory. Math. Notes, 71 (2002), no. 2, 245–261. arXiv: math/0006046.

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Savin and B. Sternin. The eta invariant and parity conditions. Adv. in Math., 182 (2004), no. 2, 173–203.

    Article  MathSciNet  MATH  Google Scholar 

  64. A.Yu. Savin and B.Yu. Sternin. Elliptic operators in even subspaces. Matem. sbornik, 190 (1999), no. 8, 125–160. English transl.: Sbornik: Mathematics 190 (1999), no. 8, 1195–1228; arXiv: math/9907027.

    MathSciNet  Google Scholar 

  65. A.Yu. Savin and B.Yu. Sternin. Elliptic operators in odd subspaces. Matem. sbornik, 191 (2000), no. 8, 89–112. English transl.: Sbornik: Mathematics 191 (2000), no. 8, arXiv: math/9907039.

    MathSciNet  Google Scholar 

  66. B.-W. Schulze, B. Sternin, and V. Shatalov. On general boundary value problems for elliptic equations. Math. Sb., 189 (1998), no. 10, 145–160. English transl.: Sbornik: Mathematics 189 (1998), no. 10, 1573-1586.

    MathSciNet  Google Scholar 

  67. R.T. Seeley. Singular integrals and boundary value problems.Am. J. Math., 88 (1966), 781–809.

    Article  MathSciNet  MATH  Google Scholar 

  68. R.T. Seeley. Complex powers of an elliptic operator. Proc. Sympos. Pure Math., 10 (1967), 288–307.

    MathSciNet  Google Scholar 

  69. R.T. Seeley. Topics in pseudodifferential operators. In Pseudo-Differential Operators, 167–305, Roma, 1969.

    Google Scholar 

  70. I.M. Singer. Eigenvalues of the Laplacian and invariants of manifolds. Proceedings of the International Congress of Mathematicians, pages 187–199, Vancouver, 1974.

    Google Scholar 

  71. D. Sullivan. Geometric Topology. Localization, Periodicity and Galois Symmetry. MIT, Cambridge, Massachusets, 1970.

    Google Scholar 

  72. E. Witten. Global gravitational anomalies. Commun. Math. Phys., 100 (1985), 197–229.

    Article  MathSciNet  MATH  Google Scholar 

  73. K.Wojciechowski. A note on the space of pseudodifferential projections with the same principal symbol. J. Operator Theory, 15 (1986), no. 2, 207–216.

    MathSciNet  MATH  Google Scholar 

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  1. Independent University of Moscow, B. Vlasyevskiy pereulok, d. 11, 119002, Moscow, Russia

    Anton Savin & Boris Sternin

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  1. Anton Savin
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  2. Boris Sternin
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Editors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckih 8, 00-956, Warszawa, Poland

    Bogdan Bojarski

  2. Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, 119992, Moscow, Russia

    Alexander S. Mishchenko  & Evgenij V. Troitsky  & 

  3. Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02-097, Warszawa, Poland

    Andrzej Weber

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Savin, A., Sternin, B. (2006). Pseudodifferential Subspaces and Their Applications in Elliptic Theory. In: Bojarski, B., Mishchenko, A.S., Troitsky, E.V., Weber, A. (eds) C*-algebras and Elliptic Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7687-1_12

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