Can't we really add together two points on a manifold?

Question

Let us consider a classical mechanical system with observables being smooth functions $C^\infty(X)$ on a Poisson manifold $X$. The algebra of observables will be denoted as $A$

Next we can define states as linear real-valued functions on this algebra of observables, see an entry in nLab.:

A classical state is a linear function $\rho: A \to \mathbb R$

• which is positive in that for all $a \in A$ we have that
• and which is normalized in that $\rho(1) = 1$

Then it proceeds with the definition of pure state:

a pure state is a state that is not only a linear map, but even an associative algebra homomorphism $\rho : A \to \mathbb R$.

After that a remarkable statement is made:

If the classical mechanical system comes from a Poisson manifold $(X,\{-,-\})$ ... then the pure states correspond precisely to the points of the manifold $X$. So each point of $X$ is one specific (= “pure”) state that the mechanical system defined by $(X,\{-,-\})$ can be in, whereas a general state $\rho : A \to R$ is a distribution of such specific states.

Unfortunately I could not find this statement in the references provided at nLab. Could anyone please suggest any sources to substantiate the claim above?

What bothers me, is that pure states being linear functions should admit an algebra structure with observable-wise addition and summation. Does it mean there is a natural algebra structure for the points of a manifold?

I was sure, that a sum of two points should not make sense, like it was explained by Qiaochu Yuan in this answer.

As for additional sources, Peter Bongaarts's book Quantum Theory. A Mathematical Approach states that there exists

... a remarkable theorem, unfortunately in an unpublished preprint, stating that two manifolds $\mathcal M_1$ and $\mathcal M_2$ are diffeomorphic if and only if the corresponding algebras of functions $C^\infty(\mathcal M_1)$ and $C^\infty(\mathcal M_2)$ are isomorphic, and that moreover a manifold $\mathcal M$ can be reconstucted from its function algebra $C^\infty(\mathcal M)$.

The book does not explain this "reconstruction". The unpublished preprint is Thomas, E.G.F.: Characterization of a manifold by the $*$-algebra of its $C^\infty$ functions. Preprint Mathematical Institute, University of Groningen, however there is apparently no easy access to this work.

Answer 1

Yes, can you can - it's just that the sum of two "points", i.e., the sum of the pure states associated to two points, is not necessarily itself a pure state. As your excerpt says:

... whereas a general state $\rho:A\to \mathbb{R}$ is a distribution of such specific states.

So there is certainly not, in general, an addition operation on the points of the manifold that produces a point on the manifold.