Advertisement

Similarity Problems, Følner Sets and Isometric Representations of Amenable Semigroups

  • Catalin BadeaEmail author
  • Laurian Suciu
Article
  • 5 Downloads

Abstract

We revisit Sz.-Nagy’s criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier–Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Følner sets in similarity problems in both settings of unimodular, \(\sigma \)-compact, amenable groups and in discrete semigroups possessing the Strong Følner condition (SFC).

Keywords

Operators similar to isometries unitarizable representations amenable semigroups Følner conditions 

Mathematics Subject Classification

47A05 47A15 43A07 

Notes

References

  1. 1.
    Argabright, L.N., Wilde, C.O.: Semigroups satisfying a strong Følner condition. Proc. Am. Math. Soc. 18, 587–591 (1967)zbMATHGoogle Scholar
  2. 2.
    Berkson, E., Gillespie, T.A.: Mean-2-bounded operators on Hilbert space and weight sequences of positive operators. Positivity 3, 101–133 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bermúdez, T., Bonilla, A., Müller, V., Peris, A.: Cesàro bounded operators in Banach spaces. arXiv:1706.03638
  4. 4.
    van Casteren, J.A.: Operators similar to unitary or selfadjoint ones. Pac. J. Math. 104(1), 241–255 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    van Casteren, J.A.: Boundedness properties of resolvents and semigroups of operators. In: Linear Operators (Warsaw, 1994). Banach Center Publications, vol. 38, pp. 59–74. Polish Acad. Sci. Inst. Math., Warsaw (1997)Google Scholar
  6. 6.
    Day, M.M.: Means for the bounded functions and ergodicity of the bounded representations of semi-groups. Trans. Am. Math. Soc. 69, 276–291 (1950)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Day, M.M.: Amenable semigroups. Ill. J. Math. 1, 509–544 (1957)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dixmier, J.: Les moyennes invariantes dans les semi-groupes et leurs applications. Acta Sci. Math. Szeged. 12, 213–227 (1950) (Leopoldo Fejér et Frederico Riesz LXX annos natis dedicatus, Pars A)Google Scholar
  9. 9.
    Emerson, W.R.: Large symmetric sets in amenable groups and the individual ergodic theorem. Am. J. Math. 96, 242–247 (1974)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fack, T.: A Dixmier’s theorem for finite type representations of amenable semigroups. Math. Scand. 93(1), 136–160 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gray, R.D., Kambites, M.: Amenability and geometry of semigroups. Trans. Am. Math. Soc. 369, 8087–8103 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, B.Z., Zwart, H.: On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform. Integral Equ. Oper. Theory 54, 349–383 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klawe, M.: Semidirect product of semigroups in relation to amenability, cancellation properties, and strong Følner conditions. Pac. J. Math. 73, 91–106 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sz.-Nagy, B.: On uniformly bounded linear transformations in Hilbert space. Acta Univ. Szeged. Sect. Sci. Math. 11, 152–157 (1947)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nakamura, M., Takeda, Z.: Group representation and Banach limit. Tohoku Math. J. (2) 3, 132–135 (1951)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Namioka, I.: Følner’s conditions for amenable semi-groups. Math. Scand. 15, 18–28 (1964)MathSciNetCrossRefGoogle Scholar
  17. 17.
    von Neumann, J.: Zur allgemeinen theorie des maßes. Fund. Math. 13, 73–111 (1929)CrossRefGoogle Scholar
  18. 18.
    Nevo, A.: Pointwise ergodic theorems for actions of groups. In: Handbook of Dynamical Systems, vol. 1B, pp. 871–982. Elsevier, Amsterdam (2006)Google Scholar
  19. 19.
    Paterson, A.L.T.: Amenability. Mathematical Surveys and Monographs, vol. 29. American Mathematical Society, Providence (1988)zbMATHGoogle Scholar
  20. 20.
    Pisier, G.: Similarity Problems and Completely Bounded Maps. Second and Expanded Edition. Includes the Solution to the Halmos Problem. Lecture Notes in Mathematics, vol. 1618. Springer, Berlin (2001)zbMATHGoogle Scholar
  21. 21.
    Pisier, G.: Are unitarizable groups amenable? In: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol. 248, pp. 323–362. Birkhauser, Basel (2005)Google Scholar
  22. 22.
    Pruvost, B.: Analytic equivalence and similarity of operators. Integral Equ. Oper. Theory 44, 480–493 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pruvost, B.: Problèmes de similarités, opérateurs de type Foguel et calcul fonctionnel. PhD thesis, Univ. Lille (2003)Google Scholar
  24. 24.
    Tessera, R.: Large scale Sobolev inequalities on metric measure spaces and applications. Rev. Mat. Iberoam. 24(3), 825–864 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Univ. Lille, CNRS, UMR 8524-Laboratoire Paul PainlevéLilleFrance
  2. 2.Department of Mathematics and Informatics“Lucian Blaga” University of SibiuSibiuRomania

Personalised recommendations