The converse of von Neumann's mean ergodic theorem 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?
 A: If $\Gamma$ is a property (T) group with infinite conjugacy classes, e.g., $\Gamma = PSL_3(\mathbb Z)$, then $\Gamma$ will have such a sequence. The ``spectral gap'' criterion for property (T) shows that there exists $c > 0$ and $F \subset \Gamma$ finite such that for any unitary representation $\pi$ we have $\left\| \frac{1}{|F|} \sum_{\gamma \in F} \pi(\gamma) - P \right\| < 1 - c$. Since $\Gamma$ has infinite conjugacy classes a short argument shows that there exists a sequence $\{ \gamma_n \}_{n = 1}^\infty \subset \Gamma$ so that $F_n = ( \gamma_n F \gamma_n^{-1}) ( \gamma_{n-1} F \gamma_{n-1}^{-1} ) \cdots (\gamma_1 F \gamma_1^{-1} )$ satisfies $| F_n | = | F |^n$ we then have 
$$
\left\| \frac{1}{|F_n|} \sum_{\gamma \in F_n} \pi(\gamma) - P \right\| 
= \left\|  \left(\frac{1}{|F|} \sum_{\gamma \in F} \pi(\gamma_n\gamma\gamma_n^{-1}) - P\right )\cdots \left( \frac{1}{|F|} \sum_{\gamma \in F} \pi(\gamma_1\gamma\gamma_1^{-1}) - P \right)\right\| 
$$
$$
\leq (1 - c)^n \to 0.
$$
Hence, this sequence of operators also converges in the strong operator topology.
