Markov–Kakutani fixed-point theorem

In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups.

Statement

edit

Let   be a locally convex topological vector space, with a compact convex subset  . Let   be a family of continuous mappings of   to itself which commute and are affine, meaning that   for all   in   and   in  . Then the mappings in   share a fixed point.[1]

Proof for a single affine self-mapping

edit

Let   be a continuous affine self-mapping of  .

For   in   define a net   in   by

 

Since   is compact, there is a convergent subnet in  :

 

To prove that   is a fixed point, it suffices to show that   for every   in the dual of  . (The dual separates points by the Hahn-Banach theorem; this is where the assumption of local convexity is used.)

Since   is compact,   is bounded on   by a positive constant  . On the other hand

 

Taking   and passing to the limit as   goes to infinity, it follows that

 

Hence

 

Proof of theorem

edit

The set of fixed points of a single affine mapping   is a non-empty compact convex set   by the result for a single mapping. The other mappings in the family   commute with   so leave   invariant. Applying the result for a single mapping successively, it follows that any finite subset of   has a non-empty fixed point set given as the intersection of the compact convex sets   as   ranges over the subset. From the compactness of   it follows that the set

 

is non-empty (and compact and convex).

Citations

edit
  1. ^ Conway 1990, pp. 151–152.

References

edit
  • Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
  • Markov, A. (1936), "Quelques théorèmes sur les ensembles abéliens", Dokl. Akad. Nauk SSSR, 10: 311–314
  • Kakutani, S. (1938), "Two fixed point theorems concerning bicompact convex sets", Proc. Imp. Akad. Tokyo, 14: 242–245
  • Reed, M.; Simon, B. (1980), Functional Analysis, Methods of Mathematical Physics, vol. 1 (2nd revised ed.), Academic Press, p. 152, ISBN 0-12-585050-6