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K-Theory and Perturbations of Absolutely Continuous Spectra

  • Dan-Virgil Voiculescu
Article

Abstract

We study the K0-group of the commutant modulo a normed ideal of an n-tuple of commuting Hermitian operators in some of the simplest cases. In case n = 1, the results, under some technical conditions are rather complete and show the key role of the absolutely continuous part when the ideal is the trace-class. For a commuting n-tuple, \({n \ge 3}\) and the Lorentz (n,1) ideal, we show under an absolute continuity assumption that the commutant determines a canonical direct summand in K0. Also, certain properties involving the compact ideal, established assuming quasicentral approximate units mod the normed ideal, have weaker versions which hold assuming only finiteness of the obstruction to quasicentral approximate units.

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References

  1. 1.
    Bercovici H., Voiculescu D.V.: The analogue of Kuroda’s theorem for n-tuples. The Gohberg anniversary collection, vol. II. Oper. Theory Adv. Appl. 41, 57–60 (1989)zbMATHGoogle Scholar
  2. 2.
    Blackadar B.: K-Theory for Operator Algebras, vol. 5. MSRI Publications, Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bourgain, J., Voiculescu, D. : The essential centre of the mod-a-diagonalization ideal commutant of an n-tuple of commuting Hermitian operators. arXiv:1309.2145
  4. 4.
    Carey R.W., Pincus J.D.: Perturbation by trace-class operators. Bull. AMS 80(4), 758–759 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Connes A.: On the spectral characterization of manifolds. J. Noncommut. Geom. 7(1), 1–82 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Higson N., Roe J.: Analytic K-Homology. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  7. 7.
    Gohberg I.C., Krein M.G.: Introduction to the Theory of Linear Non-self-Adjoint Operators. Translations of Mathematical Monographs, vol. 18. AMS, Providence (1969)Google Scholar
  8. 8.
    Kato T.: Perturbation Theory for Linear Operators. Classics of Mathematics. Springer, Berlin (1995)CrossRefGoogle Scholar
  9. 9.
    Martin M., Putinar M.: Lectures on Hyponormal Operators. Operator Theory: Advances and Applications, vol. 39. Birkhauser, Berlin (1989)CrossRefGoogle Scholar
  10. 10.
    Reed M., Simon B.: Methods of Modern Mathematical Physics, Vol. III: Scattering Theory. Academic Press, London (1979)zbMATHGoogle Scholar
  11. 11.
    Simon B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120, 2nd edn. AMS, Providence (2005)Google Scholar
  12. 12.
    Voiculescu D.V.: Some results on norm-ideal perturbations of Hilbert space operators I. J. Oper. Theory 1, 3–37 (1979)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Voiculescu D.V.: Some results on norm-ideal perturbations of Hilbert space operators II. J. Ope. Theory 5, 77–100 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Voiculescu D.: On the existence of quasicentral approximate units relative to normed ideals. Part I. J. Funct. Anal. 91, 1–36 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Voiculescu D.V.: Perturbation of Operators, Connections with Singular Integrals, Hyperbolicity and Entropy. Harmonic Analysis and Discrete Potential Theory (Frascati, 1991), pp. 181–191. Plenum Press, New York (1992)Google Scholar
  16. 16.
    Voiculescu D.V.: Almost normal operators mod Hilbert–Schmidt and the K-theory of the algebras \({E\Lambda(\Omega)}\) . J. Noncommut. Geom. 8(4), 1123–1145 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Voiculescu D.V.: Countable degree −1 saturation of certain \({C^*}\) -algebras which are coronas of Banach algebras. Groups Geom. Dyn. 8, 985–1006 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Voiculescu D.V.: Some \({C^*}\) -algebras which are coronas of non-\({C^*}\) -Banach algebras. J. Geom. Phys. 105, 123–129 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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