# smooth or differential version of Gelfand-Neimark theorem

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?

• One can reconstruct a smooth manifold (not necessarily compact) from the algebra of smooth functions on it. This is the point of view explored systematically in Smooth Manifolds and Observables by "Jet Nestruev". – Lord Shark the Unknown Apr 26 '18 at 16:59
• Ok @LordSharktheUnknown. But given an arbitray commutative C*-algebra with unity, how can I see that this algebra is or not "like" an algebra of smooth functions for some manifold? This question is because I would like to define a subcategory on commC*-alg1 – GaSa Apr 26 '18 at 17:30
• I'm following this thread because it's since I was taught Gel'fand duality the first time that I wonder how far can it be extended! – Fosco Loregian Apr 26 '18 at 19:42