I am trying to recover a Poisson manifold from its algebraic (commutative) dual. So I looked at the side of Gelfand transformation (since a Poisson algebra is a commutative Banach algebra). Gelfand transformation states exactly this:
$A \to C(Spec(A)), a \mapsto (h \to h (a))$
My question is how to precisely determine $h$. We should normally find the characters of A (by Casimir functions ?). But after ? I can not conclude.
Thanks for your help
Nicolas