Beyond Markov–Kakutani

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


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.


Extreme Point Compact Group Haar Measure Topological Vector Space Common Fixed Point 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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