Exercise 5, Appendix 0

Let $G$ be a discrete group. a) Show that a mean can be defined on $G$ (exerc. 4) if and only if the closed vector subspace $N$ of $\ell^\infty_\mathbf{R}(G)$ generated by the functions $\gamma(s)\ x - x$, where $s \in G$ and $x \in \ell^\infty_\mathbf{R}(G)$, is distinct from $\ell^\infty_\mathbf{R}(G)$ (use the Hahn-Banach theorem). b) Suppose that for every $\varepsilon > 0$ and for every finite sequence $s_1, ..., s_k$ of elements of $G$, there exists a non-empty finite subset $F$ of $G$ such that

Show that a mean can be defined on $G$ (use $a$) and show that $1 \notin N$. c) Suppose a mean can be defined on $G$. Let $\varepsilon > 0$ and let $s_1, ..., s_k$ be elements of $G$. Show that there exists a vector $x \geq 0$ in the space $\ell^1_\mathbf{R}(G)$ such that $\|x\| = 1$ and that

where $x$ ranges over the set of all vectors of $\ell^1_\mathbf{R}(G)$ such that $x \geq 0$ and $\|x\| = 1$. Show that 0 belongs to the closure of the convex set $C$ for the topology $\sigma(E, E')$; for this, use $a$, observing that the unit ball of $E$ is dense in the unit ball of $E''$ for $\sigma(E'', E')$. d) For every $x \geq 0$ in $\ell^1_\mathbf{R}(G)$, and for every $a > 0$, let $x_a$ denote the characteristic function of the set of all $s \in G$ such that $x(s) \geq a$ (i.e. $x_a(s) = 1$ if $x(s) \geq a$, $x_a(s) = 0$ if $x(s) < a$). Then for every $s \in G$,

e) Show that, if a mean can be defined on $G$, then for every $\varepsilon > 0$, and for every finite sequence of elements $s_1, ..., s_k$ of $G$, there exists a non-empty finite subset $F$ of $G$ such that

(Show that, having chosen $x$ as in $c$), there exists an $a > 0$ such that $\sum_{j=1}^k \| \gamma(s_j)\ x_a - x_a \| \leq \varepsilon$; use $d$). \*