Definitions
Let $G$ be a Lie group and $\mu$ be a Haar measure on $G$.
A compact-convex $G$-space is a $G$-invariant compact and convex subset of a locally convex topological vector space $V$ on which $G$ acts continuously (i.e., the map $G\times V\to V$ is continuous).
We say that $G$ is amenable if whenever there is a compact-convex $G$-space $K$, there is a point $v$ in $K$ which is fixed by each member of $G$.
Let $V$ be a linear subspace of $L^\infty(G, \mu)$ which contains the constant function $1_G$.
A mean on $V$ is a linear map $\lambda:V\to \mathbb C$ such that
$\bullet$ $\lambda(1_G)=1$, and
$\bullet$ $\lambda(f)\geq 0$ whenever $f\geq 0$.
Question
Proposition 12.3.9 in Dave Witte Morris's Arithmetic Groups is the following:
Proposition. If $G$ is amenable then there is a left-invariant mean on the space $C_b(G)$ of all the bounded continuous functions on $G$.
The proof provided in the text is as follows:
Let $\mathcal M$ be the set of all the means on $C_b(G)$. Note that $\mathcal M$ is a non-empty, convex, and left invariant subset of $C_b(G)^*$. Further, $\mathcal M$ is a weak* closed subset of the unit ball in $C_b(G)^*$. Therefore, by the Banach-Alaoglu theorem, we have that $\mathcal M$ is compact.
At this point the author says that the amenability of $G$ now implies the existence of a left-invariant mean.
However, the author himself points out that there is a technical difficulty here since it is not clear that the action of $G$ on $\mathcal M$ is continuous.
So my question is how do we get around this difficulty?
The problem here is that $G$ doesn't act continuously on $\mathcal{C}^b(G)$ with respect to the supremum norm. In fact, the fixed-point property you defined doesn't imply a mean on the set of bounded continuous functions on $G$ but a mean on the set $\mathcal{C}_{ru}^b(G)$ of bounded right-uniformly continuous functions on $G$ ( and a mean on this set implies one on $\mathcal{C}^b(G)$ but to show this you have to work a little ).
For instance $\mathcal{C}_{ru}^b(G)$ is composed by all the elements $f$ of $\mathcal{C}^b(G)$ such that the orbital map $g \longmapsto gf$ is continuous for the supremum topology. Consequently, the adjoint action on the dual will be weak-* continuous and so you can use the fixed-point property.
I hope it is clear :-)