The fixed point property implies amenability

Put $X:=\mathrm{U}\mathrm{C}\mathrm{B}(G)$ and $E:=X^{\ast }$ with the weak-star topology. By [F3], every evaluation on $E$ is continuous and linear. Therefore addition and scalar multiplication on $E$ are continuous, since their evaluations are the corresponding sums and scalar multiples. The basic zero-neighborhoods are finite intersections of inverse images of open disks under linear evaluations; these neighborhoods are convex. Distinct functionals differ on some $\psi \in X$; disjoint scalar neighborhoods of their evaluations pull back to disjoint weak-star neighborhoods, so $E$ is Hausdorff. Hence $E$ is a Hausdorff locally convex topological vector space by [F4].

Let $M$ be the set of positive complex-linear functionals $m$ on $X$ with $m(1_G)=1$. It is nonempty because evaluation ${\delta }_e(\psi )=\psi (e)$ is a mean, and it is convex. Every real-valued $u\in X$ satisfies $m(u)\in \mathbb{R}$ and $\mathrm{\mid }m(u)\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$: positivity applied to $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}{1}_G+u$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}{1}_G-u$ gives both claims. Real and imaginary parts of UCB functions remain in $X$, since their translation differences are bounded by the original difference. If $m(\psi )\mathrm{\ne }0$, set $\alpha =\overline{m(\psi )}\mathrm{/}\mathrm{\mid }m(\psi )\mathrm{\mid }$. Then $\mathrm{\mid }\alpha \mathrm{\mid }=1$ and $m(\mathrm{Re}(\alpha \psi ))=\mathrm{\mid }m(\psi )\mathrm{\mid }$; also $\mathrm{Re}(\alpha \psi )\le \mathrm{\mid }\psi \mathrm{\mid }\le \mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}{1}_G$. Positivity gives $\mathrm{\mid }m(\psi )\mathrm{\mid }\le \mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$, and the same bound is immediate if $m(\psi )=0$. Thus $M\subseteq B_{X^{\ast }}$.

The set $M$ is weak-star closed: it is the intersection of $\{m:m(1_G)=1\}$ and, for every nonnegative $\psi \in X$, $\{m:m(\psi )\in [0,\mathrm{\infty })\}$; these are closed by [F3]. By [A1] and [F5], the ultrafilter lemma holds, so [F6] makes $B_{X^{\ast }}$ compact. Since $M$ is a closed subset, [F7] makes $M$ compact. Together with step 1.2, $M$ is a nonempty compact convex subset of the locally convex space $E$.

For $g\in G$ and $m\in M$, define $(g\cdot m)(\psi ):=m(L_{g^{-1}}\psi )$ for $\psi \in X$. Translation invariance of $X$ shows this is well-defined; positivity and $L_{g^{-1}}{1}_G=1_G$ show $g\cdot m\in M$. The identity $L_a{L}_b=L_{ab}$ gives $g\cdot (h\cdot m)=(gh)\cdot m$, and linearity in $m$ makes each map $m\mapsto g\cdot m$ affine.

Fix $(g_0,m_0)\in G\times M$, $\psi \in X$, and $\eta >0$. By [F1], choose a neighborhood $V$ of $g_0$ with $\mathrm{\parallel }{L}_{g^{-1}}\psi -L_{g_0^{-1}}\psi {\mathrm{\parallel }}_{\mathrm{\infty }}<\eta \mathrm{/}2$ for $g\in V$. By [F3], choose a weak-star neighborhood $W$ of $m_0$ such that $\mathrm{\mid }(m-m_0)(L_{g_0^{-1}}\psi )\mathrm{\mid }<\eta \mathrm{/}2$ for $m\in W$. For $g\in V$ and $m\in W\cap M$, step 1.2 gives $\mathrm{\mid }(g\cdot m)(\psi )-(g_0\cdot m_0)(\psi )\mathrm{\mid }\le \mathrm{\parallel }m\mathrm{\parallel }\mathrm{\parallel }{L}_{g^{-1}}\psi -L_{g_0^{-1}}\psi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\mid }(m-m_0)(L_{g_0^{-1}}\psi )\mathrm{\mid }<\eta$. Thus every evaluation of the action is continuous; by the initial weak-star topology, the action $G\times M\to M$ is continuous. It is affine by step 2.2.

The fixed point property applied to the continuous affine action of step 2.2 on the nonempty compact convex set $M$ gives a fixed point $m\in M$. Thus $m(L_{g^{-1}}\psi )=m(\psi )$ for every $g\in G$ and $\psi \in X$; as $g^{-1}$ ranges over $G$, $m$ is a left-invariant mean on $\mathrm{U}\mathrm{C}\mathrm{B}(G)$.

By [F8] and step 4.1, $G$ satisfies (P1); [F12] gives a Reiter net. AC supplies the ultrafilter lemma by [F5], so [F9] gives a left-invariant mean on $L^{\mathrm{\infty }}(G)$. By [F10], $G$ is amenable.