Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$.
But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the underlying manifold. E.g., if we consider $C^1(M)$ then there are differential operators using coordinates on $M$.
Question: what is the correct framework of "reconstructing" the underlying manifold from the spaces of functions?
I know of one approach — use maximal ideals. The space itself then is reconstructed as the space of maximal ideals. But this does not allow to reason about differential operators.
Context: in quantum mechanics, consider $n$-electron atoms for different $n$. Clearly, for larger $n$, the wave functions are defined on larger spaces. But as abstract Hilbert spaces, their state spaces are isomorphic. What else do we need to add to the Banach (or Hilbert) space to be able to reconstruct the manifold? And how much do we need to assume of the space (beyond being Banach)? Should it be done on the level of operator algebra rather than the space itself?
