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Let $G$ be an infinite countable group. If $G$ is amenable then it has a number of other interesting properties. To prove such a property from the existence of an invariant mean on $G$, we usually start with any invariant mean and use it for some kind of averaging, or exploit its invariance.

Question: Are there any proofs of properties of $G$ in which we cannot use an arbitrary invariant mean? So the proof works with some invariant means on $G$ but not with others?

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  • $\begingroup$ I'm not sure there's any precise answer, but for instance when $G$ is non-abelian, there are bi-invariant means, while probably there are also left-invariants that are not right-invariant. More generally, one can ask invariance by any amenable group of automorphisms. $\endgroup$ – YCor Feb 6 '17 at 19:19
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    $\begingroup$ @YCor Perhaps there is some ambiguity in the question because amenability is equivalent to several different types of invariance. If the application works with an arbitrary bi-invariant mean then that would not be a good answer. A more pertinent example, if there is one, would be the requirement to start with an extreme point of the convex set of invariant means, or at the opposite end with a mean that has a large "support". I didn't put that in the question, hoping that someone would come up with something more exciting. $\endgroup$ – user95282 Feb 6 '17 at 20:01

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