Exercise 4, § 1

Let G be a topological group. One says that G is amenable if there exists a linear form $\lambda :\mathscr{C}_b(G;\mathbf{R})\rightarrow \mathbf{R}$ such that

(i) If $f\geqslant 0$, then $\lambda (f)\geqslant 0$.

(ii) One has $\lambda (1) = 1$.

(iii) For every $g\in G$ and every $f\in \mathscr{C}_b(G;\mathbf{R})$, one has $\lambda (\boldsymbol{\gamma }(g)f) =\lambda (\boldsymbol{\delta }(g)f)$, where

If G is a discrete group, this definition coincides with that of EVT, IV, p. 73, exerc. 4.

a) The topological group G is amenable if and only if for every integer $n\in \mathbf{N}$, every finite family $(f_i)_{1\leqslant i\leqslant n}$ in $\mathscr{C}_b(G;\mathbf{R})$, every families $(g_i)_{1\leqslant i\leqslant n}$ and $(h_i)_{1\leqslant i\leqslant n}$ in G, every real number $a$, the condition

for every $x\in G$ implies that $a\leqslant 0$. (To prove that this condition is sufficient, consider the set of functions of the form

and apply Hahn–Banach theorem.)

b) Every compact topological group is amenable, every abelian topological group is amenable.

c) Let G be a non-commutative free group. The discrete group G is not amenable.

d) For every Hilbert space E and every bounded continuous representation of G in E, there exists a Hilbert norm $q$ on E such that the representation of G in the Hilbert space E endowed with $q$ is unitary.