Let $H = \prod_{i=1}^\infty H_i$ be a product of discrete, countable, amenable groups $H_i$ and $G$ be a countable group. Let $x: H \to G$ be a function and recall that $H$ acts on the set of functions from $H$ to $G$ by $(\alpha_h x)(\tilde{h}) = x(h^{-1}\tilde{h})$
Suppose that we have $h,h'\in H$, $h= (h_i)_{i\in \mathbb{N}}$, $h'=(h'_i)_{i\in \mathbb{N}}$ such that $\alpha_h x \neq \alpha_{h'} x$. Is it true that there exists $J \in \mathbb{N}$ such that $\alpha_{h_1\dots h_J}x \neq \alpha_{h'_1 \dots h'_J} x$?
I suppose that it could be false but I don't have a counterexample. Moreover, I don't know if there is a general theory that studies these things.
Thanks in advice.
Well, I think I can self-answer my question: Take $H$ as above and $G=\mathbb{Z}/2\mathbb{Z}$,
$x(\tilde{h})= \begin{cases} 0 &\mbox{if there exists} \, J \in \mathbb{N} \, \mbox{such that} \, \tilde{h}_j = 0 \, \mbox{for every}\, j>J\\ 1 &\mbox{if not}; \end{cases}$
and consider $h\in H$ any element with $x(h)=1$. Then, $(\alpha_h x)(\tilde h)=x(h^{-1}\tilde{h}) \neq x(\tilde{h})$ (for example, $1 = x(h) \neq x(1)$)
but for every projection of $h$, say $h_1\dots h_J$, we have that $$\theta_{h_1...h_J}x(\tilde h) = x(\tilde h)$$