Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi\_X:G\rightarrow U(L^2(X,\mu))$ defined by
$(\pi\_X(g)\xi)(x)=\xi(g^{-1}x)\sqrt{\chi(g^{-1},x)}$ for $g\in G$, $\xi\in L^2(X,\mu)$.


We say that the action is amenable if the trivial representation is weakly contained in $\pi\_X$. My question is that is every action amenable, whenever the group $G$ is amenable?



Amenability of the Koopman representation
Yes, see, for instance the original paper by Bekka (1990).


