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Totentanz
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Totentanz
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Isomorphic algebras determine diffeomorphic manifolds

It is a kind of folklore but I would like to see the proof of the following fact: given two smooth manifolds $M$ and $N$ if we assume that the algebras $C^{\infty}_0(M)$ and $C^{\infty}_0(N)$ are isomorphic (as algebras) then $M$ and $N$ are diffeomorphic.