For a compact manifold $M$ the space of smooth functions $C^{\infty}(M)$ is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to characterise those Fréchet subalgebras coming from a differential structure? As a naive guess, how about the following:


Conjecture: We have a bijective correspondence between smooth structures on a manifold $M$ and Fréchet algebras that are $\|-\|\_{\infty}$ norm dense in the continuous functions $C(M)$ and are maximal with respect to these properties.


Sorry if this guess if obviously wrong, but its purpose to demonstrate the type of result I am looking for.


\**Edit: I would also be interested in analogous results for topological spaces with more structure: For example, topological groups with the requirement that our Fréchet algebra contained the representable functions.*



A Fréchet space characterization of smooth structures on topological spaces?
The book *Smooth Manifolds and Observables* by the pseudonymous Jet Nestruev may be of interest, as it defines and studies smooth manifolds using only the algebra of smooth functions. However, their definition of smooth structure is somewhat disappointing, as it requires the algebra to be locally isomorphic to $C^\infty(\mathbb{R}^n)$. It would be nice to have an algebraic characterization of smooth structures that doesn't require us to already have the smooth structure on $\mathbb{R}^n$ in-hand. I don't know of a solution, but I can list some more properties one might impose on the algebra.


1. It should contain the constant function $1$.
2. It should separate points, or more strongly, satisfy an Urysohn lemma.
3. It should be local. Given a continuous function $f$, if every point has a neighborhood such that $f$ coincides with a function in the algebra on that neighborhood, then $f$ itself is in the algebra. I think this condition is redundant for compact manifolds. For non-compact manifolds, it rules out algebras like $C^\infty\_b(M)$, the bounded smooth functions. You might consider this also unnecessary, since $C^\infty\_b(M)$ has enough information to recover the smooth structure.


These first three properties give lower bounds, in the sense that "small" subalgebras of $C(M)$ tend to violate them, but they do hold for the entire $C(M)$. So you could try to look for algebras which are *minimal* with respect to these properties. On the other hand, there is an upper-bounding property:


4. Dimensionality. You can define the tangent space of a point as the space of derivations of the algebra, composed with evaluation at that point. Then you can require that the tangent space of every point have full dimension $n$. (By derivation I mean a linear map from the algebra to itself which satisfies the Leibniz rule. In this sense $C(M)$ has no nontrivial derivations.)


