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?