Families of Homomorphisms in Non-commutative Gelfand Theory: Comparisons and Examples

  • Harm BartEmail author
  • Torsten Ehrhardt
  • Bernd Silbermann
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


In non-commutative Gelfand theory, families of Banach algebra homomorphisms, and particularly families of matrix representations, play an important role.D epending on the properties imposed on them, they are called sufficient, weakly sufficient, partially weakly sufficient, radical-separating or separating.I n this paper these families are compared with one another. Conditions are given under which the defining properties amount to the same. Where applicable, examples are presented to show that they are genuinely different.


Banach algebra homomorphism matrix representation sufficient family weakly sufficient family partially weakly sufficient family radicalseparating family separating family polynomial identity algebra spectral regularity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AL.
    S.A. Amitsur, J. Levitzky, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463.MathSciNetzbMATHCrossRefGoogle Scholar
  2. BES94.
    H. Bart, T. Ehrhardt, B. Silbermann, Logarithmic residues in Banach algebras, Integral Equations and Operator Theory 19 (1994), 135-152.MathSciNetzbMATHCrossRefGoogle Scholar
  3. BES04.
    H. Bart, T. Ehrhardt, B. Silbermann, Logarithmic residues in the Banach algebra generated by the compact operators and the identity, Mathematische Nachrichten 268 (2004), 3-30.MathSciNetzbMATHCrossRefGoogle Scholar
  4. BES12.
    H. Bart, T. Ehrhardt, B. Silbermann, Spectral regularity of Banach algebras and non-commutative Gelfand theory, in: Dym et al. (Eds.), The Israel Goh-berg Memorial Volume, Operator Theory: Advances and Applications, Vol. 218, Birkhäuser 2012, 123-154.Google Scholar
  5. GM.
    W.T. Gowers, B. Maurey, The unconditional basic sequence problem, Journal A.M.S. 6 (1993), 851-874.MathSciNetzbMATHGoogle Scholar
  6. HRS95.
    R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations, Operator Theory: Advances and Applications, Vol. 74, Birkhäuser Verlag, Basel 1995.Google Scholar
  7. HRS01.
    R. Hagen, S. Roch, B. Silbermann, C*-Algebras and Numerical Analysis, Marcel Dekker, New York, Basel 2001.zbMATHGoogle Scholar
  8. Kr.
    N.Ya. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, Operator Theory: Advances and Applications, Vol. 26, Birkhäuser Verlag, Basel 1987.Google Scholar
  9. Lev.
    J. Levitzky, A theorem on polynomial identities, Proc. Amer. Math. Soc. 1 (1950), 334-341.MathSciNetGoogle Scholar
  10. LR.
    V. Lomonosov, P. Rosenthal, The simplest proof of Burnside’s Theorem on matrix algebras, Linear Algebra Appl. 383 (2004), 45-47.MathSciNetzbMATHCrossRefGoogle Scholar
  11. P.
    T.W. Palmer, Banach Algebras and The General Theory of *-Algebras, Volume I: Algebras and Banach Algebras, Cambridge University Press, Cambridge 1994.Google Scholar
  12. RSS.
    S. Roch, P.A. Santos, B. Silbermann, Non-commutative Gelfand Theories, Springer Verlag, London Dordrecht, Heidelberg, New York 2011.zbMATHCrossRefGoogle Scholar
  13. Se.
    P. Šemrl, Maps on matrix spaces, Linear Algebra Appl. 413 (2006), 364-393.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Si.
    B. Silbermann, Symbol constructions and numerical analysis, In: Integral Equations and Inverse Problems (R. Lazarov, V. Petkov, eds.), Pitman Research Notes in Mathematics, Vol. 235, 1991, 241-252.Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Econometric InstituteErasmus University RotterdamRotterdamThe Netherlands
  2. 2.Mathematics DepartmentUniversity of CaliforniaSanta CruzUSA
  3. 3.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations