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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?

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Yes, see, for instance the original paper by Bekka (1990).

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    $\begingroup$ The OP's definition of amenability is different from Bekka's. Still the answer is yes: It's easy to find an approximate invariant positive vector in $L^1$ and then take the square root of it. See also [Shalom, Ann. of Math. (2) 152 (2000), Lemma 2.3] and the following remark. $\endgroup$
    – Narutaka OZAWA
    Commented May 27, 2014 at 1:44

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