# Extreme amenability of topological groups and invariant means

Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it claims (without proof) that

(the extreme amenability) is equivalent to the existence of a left invariant multiplicative mean on the space of all bounded right uniformly continuous functions on the group.

I'm not aware of any references on this result.

Question: Is there any reference on extreme amenability of topological groups and invariant means?

The action $$G\curvearrowright\beta G$$ is continuous iff $$G$$ is discrete, so for nondiscrete groups it is not true that $$G$$ is extremely amenable iff this action has a fixed point.
What one should look at instead, is the biggest compactification on which $$G$$ acts continuously, this is known as the Samuel compactification, $$S(G)$$, and can be constructed as the spectrum of the algebra $$RUCB(G)$$ of right uniformly continuous bounded functions on $$G$$. This is where the uniform continuity condition comes from. It is then true that $$G$$ is extremely amenable iff $$G\curvearrowright S(G)$$ has a fixed point. (Of course on a discrete group all continuous functions are uniformly continuous so $$\beta G$$ and $$S(G)$$ are isomorphic)
Note also that if you write explicitly what it means, for a fixed $$f\in C(G)$$, that the orbit map $$G\to C(G)$$, $$g\mapsto gf$$, is $$\|\cdot\|_\infty$$-continuous, you'll end up with a uniform continuity condition on $$f$$!
• Thanks for pointing the missing point out. I forgot to check the continuity of the action on $\beta G$, so I removed the wrong arguments from the thread. This book seems to be exactly what I'm looking for. Jun 3 at 14:49