Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.

Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ and $\pi:\Gamma\to B(H)$ be a unitary representation of $\Gamma$ on a Hilbert space $H$. Then $$\lim_{n\to\infty}\frac{1}{|F_n|}\sum_{s\in F_n} \pi(s)y=Py$$ for every $y\in H$, where $P$ is the orthogonal projection from $H$ onto $H_\Gamma=\{x\in H\,|\,\pi(s)x=x \, {\rm for\, all}\, s\in\Gamma\}.$

**Question**: Is the converse true?

More precisely, suppose a countable discrete group $\Gamma$ has a sequence of finite subsets $\{F_n\}_{n=1}^\infty$ such that for every unitary representation $\pi:\Gamma\to B(H)$, one have $$\lim_{n\to\infty}\frac{1}{|F_n|}\sum_{s\in F_n} \pi(s)y=Py$$ for every $y\in H$. Here $P$ is the orthogonal projection from $H$ onto $H_\Gamma$.

Is $\Gamma$ amenable?

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