Exercise 4, Appendix 0

Let G be a discrete group. We say that G is a group on which a mean can be defined if there exists a linear form u on $\ell^\infty_\mathbf{R}(G)$ (I, p. 4) such that $u(x) \geq 0$ for $x \geq 0$, $u(1) = 1$, and such that $u(\gamma(s)x) = u(x)$ for all $s \in G$ and all $x \in \ell^\infty_\mathbf{R}(G)$ (where $(\gamma(s)x)(t) = x(s^{-1}t)$ for all $t \in G$) (invariance under left translations). a) If we put $\tilde{x} = x(t^{-1})$ for $x \in \ell^\infty_\mathbf{R}(G)$ and $t \in G$, and if the linear form u is invariant under left translations, then the linear form $v : x \mapsto u(\tilde{x})$ is invariant under right translations, in other words $v(\delta(s)x) = v(x)$ for all $s \in G$ and for all $x \in \ell^\infty_\mathbf{R}(G)$ (where $(\delta(s)x)(t) = x(ts)$ for all $t \in G$). Put $F_x(s) = u(\delta(s)x)$ for $s \in G$ and $x \in \ell^\infty_\mathbf{R}(G)$, then $F_{\gamma(t)x}(s) = F_x(s)$ and $F_{\delta(t)x}(s) = F_x(st)$ for all $t \in G$; deduce that the linear form w on $\ell^\infty_\mathbf{R}(G)$ defined by $w(x) = v(F_x)$ is invariant under left and right translations, and is such that $w(x) \geq 0$ for $x \geq 0$ and $w(1) = 1$. \* b) Let K be a non-empty compact space and $\Gamma$ a subgroup of the group of all homeomorphisms from K onto itself. Show that if a mean can be defined on $\Gamma$, then there exists a measure $\mu \geq 0$ on K of mass 1, which is invariant under $\Gamma$. (If $a \in K$, consider the linear mapping

EXERCISES

which associates to each continuous real function $f$ in $K$, the function $\sigma \mapsto f(\sigma(a))$ belonging to $\ell^\infty_\mathbf{R}(\Gamma)$.) c) Let $E$ be a Hausdorff topological vector space over $\mathbf{R}$, and $K$ a non-empty compact convex subset of $E$. Let $\Gamma$ be a group of continuous affine transformations from $K$ onto itself. Show that if a mean can be defined on $\Gamma$, then there exists a point $b \in K$ such that $\sigma(b) = b$ for all $\sigma \in \Gamma$. (Use $b$ and consider the barycenter of $\mu$.) d) Show that a mean can be defined on a discrete group $G$ if and only if there exists a non null measure $\mu$ which is invariant under $G$, on every non-empty compact space $K$ on which $G$ operates continuously. (If $E = \overline{\mathscr{K}(G)}$, consider the unit ball $B$ in $E' = \ell^1_\mathbf{R}(G)$, endowed with $\sigma(E', E)$, and associate to each element $x \in \ell^\infty_\mathbf{R}(G) = E''$ its restriction to $B$.) If $G$ is countable, it is enough that the above property holds for every compact metrizable $K$. \*