Advertisement

Essentially Normal Operators

  • Kenneth R. Davidson
Part of the Operator Theory Advances and Applications book series (OT, volume 207)

Abstract

This is a survey of essentially normal operators and related developments. There is an overview of Weyl—von Neumann theorems about expressing normal operators as diagonal plus compact operators. Then we consider the Brown—Douglas—Fillmore theorem classifying essentially normal operators. Finally we discuss almost commuting matrices, and how they were used to obtain two other proofs of the BDF theorem.

Mathematics Subject Classification (2000)

47-02 47B15 46L80 

Keywords

Essentially normal operator compact perturbation normal diagonal almost commuting matrices extensions of C*-algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Anderson, A G*-algebra A for which Ext(A) is not a group, Ann. Math. 107 (1978), 455–458.CrossRefGoogle Scholar
  2. [2]
    W. Arveson, A note on essentially normal operators, Proc. Royal Irish Acad. 74 (1974), 143–146.zbMATHMathSciNetGoogle Scholar
  3. [3]
    W. Arveson, Notes in extensions of C*-algebras, Duke Math. J. 44 (1977), 329–355.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    M. Atiyah, K-theory, W.A. Benjamin Inc., New York, 1967.Google Scholar
  5. [5]
    H. Bercovici and D. Voiculescu, The analogue of Kuroda’s theorem for n-tuples, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), 57–60, Oper. Theory Adv. Appl. 41, Birkhäuser, Basel, 1989.Google Scholar
  6. [6]
    I.D. Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365–371.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    I.D. Berg and K. Davidson, Almost commuting matrices and a quantitative version of the Brown-Douglas-Fillmore theorem, Acta Math. 166 (1991), 121–161.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Bratteli, O., Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.zbMATHMathSciNetGoogle Scholar
  9. [9]
    L. Brown, R. Douglas and P. Fillmore, Unitary equivalence modulo the compact operators and extensions of C*-algebras, Proc. conference on Operator theory, Halifax, NS, Lect. Notes Math. 3445, Springer Verlag, Berlin, 1973.Google Scholar
  10. [10]
    L. Brown, R. Douglas and P. Fillmore, Extensions of C*-algebras and K-homology, Ann. Math. 105 (1977), 265–324.CrossRefMathSciNetGoogle Scholar
  11. [11]
    M.D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), 529–533.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    M.D. Choi and E. Effros, The completely positive lifting problem for C*-algebras, Ann. Math. 104 (1976), 585–609.CrossRefMathSciNetGoogle Scholar
  13. [13]
    K. Davidson, Almost commuting Hermitian matrices, Math. Scand. 56 (1985), 222–240.zbMATHMathSciNetGoogle Scholar
  14. [14]
    K. Davidson, Normal operators are diagonal plus Hilbert-Schmidt, J. Operator Theory 20 (1988) 241–250.zbMATHMathSciNetGoogle Scholar
  15. [15]
    K. Davidson, C*-Algebras by Example, Fields Institute Monograph Series 6, American Mathematical Society, Providence, RI, 1996.Google Scholar
  16. [16]
    R. Douglas, C*-algebra extensions and K-homology, Annals Math. Studies 95, Princeton University Press, Princeton, N.J., 1980.Google Scholar
  17. [17]
    E. Effros, D. Handelman and C. Shen, Dimension groups and their affine transformations, Amer. J. Math. 102 (1980), 385–402.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    G. Elliott, On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras, J. Algebra 38 (1976), 29–44.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    G. Elliott, On totally ordered groups and Ko, Proc. Ring Theory conf., Waterloo, D. Handelman and J. Lawrence (eds.), Lect. Notes Math. 734 (1978), 1–49, Springer-Verlag, New York, 1978.Google Scholar
  20. [20]
    R. Exel and T. Loring, Almost commuting unitary matrices, Proc. Amer. Math. Soc. 106 (1989), 913–915.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    R. Exel and T. Loring, Invariants of almost commuting unitaries, J. Funct. Anal. 95 (1991), 364–376.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    P. Friis and M. Rordam, Almost commuting self-adjoint matrices—a short proof of Huaxin’s theorem, J. reine angew. Math. 479 (1996), 121–131.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    P. Friis and M. Rordam, Approximation with normal operators with finite spectrum, and an elementary proof of the Brown—Douglas—Fillmore theorem, Pacific J. Math. 199 (2001), 347–366.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    U. Haagerup and S. Thorbjörnsen, A new application of random matrices: Ext\( (C_{red}^* (\mathbb{F}_2 )) \) is not a group, Ann. Math. 162 (2005), 711–775.CrossRefzbMATHGoogle Scholar
  25. [25]
    D. Hadwin, An operator-valued spectrum, Indiana Univ. Math. J. 26 (1977), 329–340.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    P.R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70 (1963), 241–247.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [28]
    P.R. Halmos, Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), 67–76.Google Scholar
  29. [29]
    M. Hastings, Making almost commuting matrices commute, Coram. Math. Phys. 291 (2009), no. 2, 321–345.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    G. Kasparov, The operator K-functor and extensions of C*-algebras (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 44, (1980), 571–636.zbMATHMathSciNetGoogle Scholar
  31. [31]
    T. Kato, Perturbation of continuous spectra by trace class operators, Proc. Japan Acad. 33 (1957), 260–264.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    S. Kuroda, On a theorem of Weyl-von Neumann, Proc. Japan Acad. 34 (1958), 11–15.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    H. Lin, Almost commuting Hermitian matrices and applications, Fields Institute Comm. 13 (1997), 193–233.Google Scholar
  34. [34]
    T. Loring, K-theory and asymptotically commuting matrices, Canad. J. Math. 40 (1988), 197–216.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    W. Luxemburg and R. Taylor, Almost commuting matrices are near commuting matrices, Indag. Math. 32 (1970), 96–98.MathSciNetGoogle Scholar
  36. [36]
    C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal. 33 (1979), 332–338.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [37]
    M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math. 7 (1957), 997–1010.zbMATHMathSciNetGoogle Scholar
  38. [38]
    P. Rosenthal, Research Problems: Are Almost Commuting Matrices Near Commuting Matrices?, Amer. Math. Monthly 76 (1969), 925–926.CrossRefMathSciNetGoogle Scholar
  39. [39]
    W. Sikonia, The von Neumann converse of Weyl’s theorem, Indiana Univ. Math. J. 21 (1971/1972), 121–124.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    D. Voiculescu, A non-commutative Weyl—von Neumann Theorem, Rev. Roum. Pures Appl. 21 (1976), 97–113.zbMATHMathSciNetGoogle Scholar
  41. [41]
    D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), 3–37.zbMATHMathSciNetGoogle Scholar
  42. [42]
    D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, II, J. Operator Theory 5 (1981), 77–100.zbMATHMathSciNetGoogle Scholar
  43. [43]
    D. Vioculescu, Remarks on the singular extension in the C*-algebra of the Heisenberg group, J. Operator Theory 5 (1981), 147–170.MathSciNetGoogle Scholar
  44. [44]
    D. Vioculescu, Asymptotically commuting finite rank unitaries without commuting approximants, Acta Sci. Math. (Szeged) 45 (1983), 429–431.MathSciNetGoogle Scholar
  45. [45]
    J. von Neumann, Charakterisierung des Spektrums eines Integraloperators, Actualités Sci. Indust. 229, Hermann, Paris, 1935.Google Scholar
  46. [46]
    H. Weyl, Über beschränkte quadratische Formen deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Kenneth R. Davidson
    • 1
  1. 1.Pure Math. Dept.University of WaterlooWaterlooCanada

Personalised recommendations