Day's fixed point theorem

Remarks

Suppose $G$ has the fixed point property. Let $K$ be the weak-$*$ compact convex set of means on $\ell^\infty(G)$, with the action $(g \cdot m)(f) = m(g^{-1} \cdot f)$. A fixed point is an invariant mean. Conversely, let $m$ be an invariant mean and fix $x_0 \in K$. For each continuous affine function $\varphi$ on $K$, the bounded function $g \mapsto \varphi(g \cdot x_0)$ has a mean. In a locally convex space this defines a barycenter $x \in K$ with $\varphi(x) = m(g \mapsto \varphi(g \cdot x_0))$. The invariance of $m$ makes $x$ fixed.

Example: for an abelian group, the theorem follows from the Markov–Kakutani fixed-point theorem. For the free group $F_2$ the fixed point property fails, since the set of means on $\ell^\infty(F_2)$ has no fixed point.

For a locally compact group, the same equivalence holds when the action $G \times K \to K$ is jointly continuous.

Going deeper

For a student

Day's theorem is a standard tool to produce invariant objects for actions of amenable groups. Examples are invariant probability measures for continuous actions on compact spaces, obtained by taking $K$ to be the probability measures on the space. For a non-amenable group, some action on a compact space has no invariant probability measure, and this is one way to detect non-amenability.