Assume $M$ is a compact smooth manifold (without boundary). What can we say about the spectrum of the $\mathbb{R}$-algebra $A=C^{\infty}(M)$? The elements of $M$ give rise to rational points of $A$, are there other ones? Does the smooth structure of $M$ endow $A$ with additional structure such that $M$ can be completely recovered from $A$? In other words, is there some kind of smooth Gelfand-duality?



smooth Gelfand-duality
Perhaps I should post this as an answer (even if I don't really know that theory): in 


Juan A. Navarro González & Juan B. Sancho de Salas, C∞-Differentiable Spaces, LNM 1824 [Google Books Preview](http://books.google.it/books?id=79bCwXbVnHAC&printsec=frontcover&dq=C-infinity+differentiable+spaces&source=bl&ots=xjtnnn8bSq&sig=BnSizsqq1t7wT8deQfeE--Ts4Ps&hl=it&ei=8w_DS92DFo7U7APZiu2tCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage&q&f=false)


a theory of "$C^{\infty}$ -differentiable spaces" is developped, and it would be something like the smooth analog to (possibly singular and nonreduced) complex analytic spaces.


