Isomorphisms of algebras of smooth functions revisited.

*(English)*Zbl 1082.46020Let \(M_1,M_2\) 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.

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.

Reviewer: T.S.S.R.K. Rao (Bangalore)

##### MSC:

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 |