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?



The converse of von Neumann's mean ergodic theorem
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.


