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Isomorphisms of algebras of smooth functions revisited. (English) Zbl 1082.46020
Let \(M_1,M_2\) be Hausdorff smooth finite-dimensional manifolds. Let \(C^{\infty}(M_i,F)\) denote the algebra of smooth functions over the real or complex scalar field \(F\). In this interesting paper, the author gives a proof of a Banach-Stone type theorem which describes an algebra isomorphism between these spaces as implemented by a pull back by a smooth diffeomorphism between the manifolds. It is not assumed that the manifolds are second countable or paracompact.
The canonical approach is to identify ideals of co-dimension one (or, equivalently, multiplicative functionals) with points of the manifold. Unlike this approach, the author calls an ideal of co-dimension one ‘distinguished’ if it has an element which is not in any other ideal of co-dimension one. With this notation, one of the main steps of the proof is to establish that \(p \rightarrow \{f : f(p) = 0\}\) is a homeomorphism when the set of distinguished ideals of co-dimension one is equipped with the Stone topology. It is also noted that J. Mrčun [Proc. Am. Math. Soc. 133, No. 10, 3109–3113 (2005; Zbl 1077.58005)] also gave a proof of Theorem 1, using ‘characteristic sequences of functions’ instead of characterizations of multiplicative functionals.

46E25 Rings and algebras of continuous, differentiable or analytic functions
58A05 Differentiable manifolds, foundations
58C25 Differentiable maps on manifolds
54C40 Algebraic properties of function spaces in general topology
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