This question is inspired from mathematics of quantum mechanics.
In quantum mechanics, we start with a Kähler manifold $\mathcal{M}$ (which is $\mathbb{CP}^n$), with the symplectic form $\omega$ and metric $g$.
We have a correspondence between the geometric data and algebraic data as follows. The symplectic form induces a Poisson algebra on the functions $x,y$ on the manifold $$[x,y] := \omega (dx,dy)\,,$$ and the metric induces a Jordan algebra $$x\cdot y := g(dx,dy)+xy\,.$$ Or, we can treat the two structures at the same time and say that the Kähler structure (metric+symplectic) induces a C*-algebra (Poisson+Jordan).
What seems remarkable to me is the representation of the C*-algebra. The representation is a Hilbert space $\mathcal{H}$ by GNS construction. Magically, its projective space turns out to be the manifold we started with: $$\mathbb{P}\mathcal{H} = \mathcal{M}.$$
Somehow, we go back from algebraic data to geometric data by an application of representation theory. Why is this true for $\mathcal M=\mathbb{CP}^n$? Is it true for any Kähler manifold $\mathcal{M}$? Answers and references are both appreciated.