What is the name of this construction of quantum things?
Let $E$ be a set, and let $\mathcal{F}$ be a set of maps $E \rightarrow \mathbb{R}$ that "carries all the information on $E$", that is, the map $x \in E \mapsto (f(x))_{f \in \mathcal{F}}$ is injective.
Let us call "classical things" the elements of $E$. Let us call "a representation of quantum things" a Hilbert space $\mathcal{H}$, and a map $\widehat{ \cdot } : \mathcal{F} \rightarrow Obs(\mathcal{H})$, where we denote by $Obs(\mathcal{H})$ the set of self-adjoint operators on $\mathcal{H}$, such that the map $\hat{ \cdot }$ satisfies some conditions.
Example 1: Let $E$ be $\mathbb{R}^3$, and let $\mathcal{F}$ be the set of maps $w \mapsto \langle v,w\rangle$ for $w \in \mathbb{R}^3$. Then $\mathcal{F}$ carries all the information on $\mathbb{R}^3$ since a vector is uniquely defined by the data consisting of the scalar products with all vectors.
Then every vector $\phi$ in $\mathcal{H}$ can be seen as a state of a "quantum vector", to be thought as having coordinate $a$ in the $v$-direction if, when measuring the $\widehat{( w\mapsto \langle v,w\rangle)}$ observable, one gets $a$.
As a particular case, spin $\frac{1}{2}$ can be considered as a kind of a "quantum vector with all coordinates that whenever measured, give a result in ${\pm \frac{1}{2}}$".
Example 2: the phase space of a classical $1$-dimensional particle is $\mathbb{R}\times \mathbb{R}$, and one can consider the set of maps carrying all the information to be $\mathcal{F} := {q,p} = {(x,y) \mapsto x, (x,y) \mapsto y}$.
To this classical representation corresponds the quantum representation where $\hat{q} := Q$ is the position operator, and $\hat{p} := P$ is the momentum operator.
My question is the following: I think this definition probably already exists, maybe in categorical terms, maybe in the particular case where $E$ is a manifold with enough structure. I would like to know the name of this construction; that is, the name of the general abstract framework where the two examples fit. 669