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On isomorphisms of algebras of smooth functions. (English) Zbl 1077.58005

It is shown that for any isomorphism \( T : C^\infty(N)\rightarrow C^\infty(M) \) between the algebras of smooth functions on two Hausdorff smooth manifolds \( M \) and \( N \) (not necessarily second-countable, paracompact or connected), there exists a unique diffeomorphism \( \tau : M\rightarrow N \) such that \( T(f)=f\circ\tau \) for any \( f \in C^\infty(N) \).
The same property holds for isomorphisms between the algebras of smooth functions with compact support.

MSC:

58C25 Differentiable maps on manifolds
58A05 Differentiable manifolds, foundations
46E25 Rings and algebras of continuous, differentiable or analytic functions
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