Advertisement

Beyond Markov–Kakutani

The Ryll–Nardzewski fixed-point theorem
  • Joel H. Shapiro
Chapter
Part of the Universitext book series (UTX)

Overview

In the last chapter we extended the Markov–Kakutani Theorem—originally proved only for commuting families of continuous affine maps—to “solvable” families of such maps. We used our enhanced theorem to show that every solvable group is amenable and that Haar measure exists for every topological group that is both solvable and compact. By contrast, we’ve seen (Chap.  11) that the group\(\mathcal{R}\) of origin-centric rotations of\(\mathbb{R}^{3}\) is paradoxical, hence not amenable, and therefore not solvable. Now\(\mathcal{R}\) is naturally isomorphic to the group\(\mathrm{SO(3)}\) of 3 × 3 orthogonal real matrices with determinant 1 (Appendix D), a group easily seen to be compact. Thus not every compact group is amenable.

Keywords

Extreme Point Compact Group Haar Measure Topological Vector Space Common Fixed Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 8.
    Banach, S.: On Haar’s measure. In: Theory of the Integral, by Stanisław Saks, Monografie Matematyczne, Warsaw, vol. 7, pp. 314–319 (1937) Reprinted by Dover Pub. 1964 & 2005Google Scholar
  2. 29.
    Diestel, J., Spalsbury, A.: The Joys of Haar Measure. Graduate Studies in Mathematics, vol. 150. American Mathematical Society, Providence (2014)Google Scholar
  3. 35.
    Dugundji, J., Granas, A.: A proof of the Ryll-Nardzewski fixed point theorem. J. Math. Anal. Appl. 97, 301–305 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 36.
    Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic, New York (1972)zbMATHGoogle Scholar
  5. 42.
    Haar, A.: Der Masbegriff in der Theorie der Kontinuierlichen Gruppen. Ann. of Math (2), 34, 147–169 (1933)Google Scholar
  6. 45.
    Hahn, F.: A fixed-point theorem. Math. Syst. Theory 1, 55–57 (1968)CrossRefzbMATHGoogle Scholar
  7. 49.
    Hawkins, T.: Weyl and the topology of continuous groups. In: James, I.M. (ed.) History of Topology. North Holland, Amsterdam (1999)Google Scholar
  8. 82.
    Namioka, I., Asplund, E.: A geometric proof of Ryll-Nardzewski’s fixed point theorem. Bull. Amer. Math. Soc. 73, 443–445 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 102.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  10. 103.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  11. 105.
    Ryll-Nardzewski, C.: On fixed points of semigroups of endomorphisms of linear spaces. In: Proceeding of the Fifth Berkeley Sympos Math. Statistics & Probability, pp. 55–61 (1967)Google Scholar
  12. 123.
    Willard, S.: General Topology. Addison Wesley, Reading (1970). Reprinted by Dover Publications, Mineola (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

Personalised recommendations