So I'd seen a bit of amenability from the functional-analytic point of view, but I was looking for some motivation. I found the following set of notes which I think is doing quite a nice job see here. These notes however are mostly looking at discrete groups and I was interested in the more general setting as well. I've sort of assumed that if you assume the right kind of continuity on your group action everything still works, and of course I ended up breaking maths. Specifically, here are some statements I'd seen previously:
- Abelian groups are amenable.
- Compact topological groups are amenable.
- Extenstions of amenable groups by amenable groups remain amenable.
In particular, looking at the group of isometries of $\mathbb{R}^3$ -aka $SO(3,\mathbb{R})\rtimes\mathbb{R}^3$- should be amenable as $SO(3,\mathbb{R})$ is compact and $\mathbb{R}^3$ is abelain.
That however doesn't sit quite right with Prop 1.14 from [loc. cit.], which shows if an amenable group $G$ acts on a set $X$ then $X$ is not $G$-paradoxical - but of course we know $\mathbb{R}^3$ is $SO(3,\mathbb{R})\rtimes\mathbb{R}^3$ - paradoxical by (a version of) Banach - Tarski.
As far as I can tell, the problem has to do with topology. I have a huntch the above proposition refers to discrete groups, and indeed is then proving the discrete group $SO(3,\mathbb{R})\rtimes\mathbb{R}^3$ is not amenable, while the group $SO(3,\mathbb{R})\rtimes\mathbb{R}^3$ endowed with its canonical topology is still amenable. That being said, I'm now trying to figure out where discrenteness is used. The main question I think is:
If I assume $G$ is an amenable (locally compact Haudorff) topological group, with a continuous action $G\times X \to X$ then is it true that $X$ is not $G$-paradoxical?
If that is the case, is it the case that the action of $SO(3,\mathbb{R})\rtimes\mathbb{R}^3$ on $\mathbb{R}^3$ is not jointly continuous (and is that why the proposition fails?). Also what happens with just $SO(3,\mathbb{R})$ then? Is the action of $SO(3,\mathbb{R})$ on the unit ball $\mathbb{B}^3$ also not jointly continuous? - because I think the sphere $\mathbb{S}^2$ is also $SO(3,\mathbb{R})$-paradoxical (as proved in the above notes).
Would very much appreciate some clarifications. Any recommendations of places where I could read more about this (in the general locally compact setting preferably) would also be very much appreciated. Thanks!