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.)