Using the Gelfand-Naimark theorem we can define an equivalence between the category of compact Hausdorff spaces and the category of commutative C*-algebras with unity (commC*-alg1). Someone knows if there exist a differential version? I mean, there exist an equivalence between the category of compact smooth manifolds and some subcategory of commC*-alg1?.
If the answer is "yes", exactly, who is this subcategory?