Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and 
$$\sum\_{g,h\in G}f(g)\overline{f(h)}\gamma(h^{-1}g)\geq0$$
whenever $f:G\to\mathbb{C}$ is finitely supported).
Let $(F\_N)\_{N\in\mathbb{N}}$ be a Folner sequence in $G$ and assume the Cesaro limit
$$L:=\lim\_{N\to\infty}\frac1{|F\_N|}\sum\_{g\in F\_N}\gamma(g)$$
exists. Can $L$ be negative?


If $G$ is abelian then one can apply Bochner-Herglotz theorem to represent $\gamma$ as the Fourier transform of a positive measure, decompose it into atomic and continuous components, and then apply Wiener's lemma to the continuous component.
It follows that $L\geq0$.
Can this proof be adapted to the amenable case?



Can the Cesaro limit of a positive definite function be negative?
I hope it's not frowned upon to answer one's own question, but since I just figured out the answer it doesn't make sense to keep it unanswered and there are a few upvotes so the answer may interest other people as well. 


The answer is no, i.e. $L$ must be non-negative.
The proof is quite short if one uses the right theorems: 
A version of Naimark's dilation theorem (Theorem 5.20 in [these notes of V. Paulsen](http://www.math.uh.edu/~vern/rkhs.pdf); the version [stated on wikipedia](http://en.wikipedia.org/wiki/Naimark%27s_dilation_theorem) is significantly heavier) states that there exists a Hilbert space $H$, a unitary representation $(U\_g)\_{g\in G}$ of $G$ on $H$ and a vector $v\in H$ such that 
$$\gamma(g)=\langle U\_gv,v\rangle.$$
Then the [mean ergodic theorem](http://en.wikipedia.org/wiki/Ergodic_theory#Mean_ergodic_theorem) (which holds for any Folner sequence in an amenable group) gives
$$L=\lim\_{N\to\infty}\frac1{|F\_N|}\sum\_{g\in F\_N}\langle U\_gv,v\rangle=\langle Pv,v\rangle=\|Pv\|^2\geq0$$
where $P:H\to H$ is the orthogonal projection onto the space of vectors fixed under each $U\_g$.


