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.
be a locally convex topological vector space, with a compact convex subset
be a family of continuous mappings of
to itself which commute and are affine, meaning that
share a fixed point.
be a continuous affine self-mapping of
define a net
is compact, there is a convergent subnet in
is a fixed point, it suffices to show that
(The dual separates points by the Hahn-Banach theorem; this is where the assumption of local convexity is used.)
by a positive constant
On the other hand Taking
and passing to the limit as
goes to infinity, it follows that Hence The set of fixed points of a single affine mapping
is a non-empty compact convex set
by the result for a single mapping.
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
ranges over the subset.
it follows that the set is non-empty (and compact and convex).