Recently I'm reading the paper [Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups](https://arxiv.org/pdf/math/0004010.pdf) by Pestov. When it comes to the definition of an extremely amenable topological group, it claims (without proof) that



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


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


Thanks for any comments or answers!



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 $f$!


Pestov wrote a whole book on the subject, called *Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon*, it is a great introduction to extreme amenability which contains all the details of the above and much more


