Characterization of Fréchet algebras \(C^ \infty (X)\) (Q1924566)

Let \(C^\infty(X)\) be the Fréchet \(R\)-algebra of all real-valued smooth functions on a \(\sigma\)-compact smooth \(n\)-dimensional manifold \(X\). In this paper, a characterization of these algebras among Fréchet \(R\)-algebras with locally Euclidean spectrum of dimension \(n\) is given. A characterization of algebras of \(C^\infty\)-functions on open sets in \(\mathbb{R}^n\) as separable Fréchet \(R\)-algebras with \(n\)-dimensional locally Euclidean spectrum satisfying some conditions is given as well.