1) Consider the algebra $\mathcal T (M) = \bigoplus \limits \_{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with some topology, but I do not know what to choose: the topology of pointwise convergence is one option, the compact-open topology another one, but I am open to other suggestions as well.


Is it possible to recover both the topological and the differential structure of $M$ from $\mathcal T (M)$? (This would be some sort of Gelfand-Naimark in a smooth setting; in fact, the component $\mathcal T ^{0, 0} (M)$ endowed with a suitable topology would lead to precisely some version of Gelfand-Naimark.)


A related question is: do we really need to consider all the types of tensors above? Given that covariant and contravariant are dual to each other (and we are working with manifolds of finite-dimension), shouldn't these two classes contain the same amount of information, either one being redundant for the issue under discussion? It seems that one could restrict to either $\bigoplus \limits \_{q \ge 0} \mathcal T ^{0, q} (M)$, or to $\bigoplus \limits \_{p \ge 0} \mathcal T ^{p, 0} (M)$.


2) I am curious about the same question but for a different algebra: if $\mathcal X (M)$ is the Lie algebra of the smooth vector fields on $M$, let $\mathcal U (M)$ be its universal enveloping algebra. Does $\mathcal U (M)$ encode all the information about $M$? Can $M$ be recovered from $\mathcal U (M)$? (This latter algebra, though, does not contain the smooth functions, so I have some doubts about it.)



Recovering a smooth manifold from its tensor fields
Look at $M=S^7$. This has 28 smooth structures, and their tangent bundles are all trivial - compare [Parallelizability of the Milnor's exotic spheres in dimension 7](https://mathoverflow.net/questions/58131/parallelizability-of-the-milnors-exotic-spheres-in-dimension-7)  . Thus their tensor algebras do not carry more information about the smooth structure than $C^\infty(M)$.


On the other hand, for a compact smooth manifold, the algebra of smooth functions is sufficient to recover the smooth structure: Every maximal ideal of this algebra is of the form $\{f\in C^\infty(M): f(x) = 0\}$ for some $x\in M$, and the topology on $M$ is recovered as the weakest one on the space of maximal ideals such that $\mathfrak{m}\mapsto [f]\in C^\infty(M)/\mathfrak{m}\cong \mathbb{R}$ is continuous for all $f\in C^\infty(M)$. So this space is a compact topological manifold with a subsheaf $C^\infty$ of the sheaf of continuous functions, and we take a chart $U\to\mathbb{R}^n$ to be smooth if all of its coordinate functions are in $C^\infty(U)$.


