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Multiplicative bijections between algebras of differentiable functions. (English) Zbl 1124.46016

Let \( M \) and \( N \) be two Hausdorff manifolds of positive dimension (not necessarily second-countable, paracompact or connected), and of class \( C^r \) with \( 1\leq r<\infty \).
The authors show that for any multiplicative bijection \( B : C^r(N)\rightarrow C^r(M) \) between the algebras of \( C^r \) functions on \( M \) and \( N \), there is a unique \( C^r \)-diffeomorphism \( \phi:M\rightarrow N \) such that \( B(g)=g\circ\phi \) for any \( g\in C^r(N) \). In particular, \( B \) is automatically an algebra isomorphism.

MSC:

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