Characterization of the spectra of certain function algebras
Let \(E\) denote a locally convex Hausdorff space. The authors study the spectrum \(\text{Hom } C^ \infty(E)\) of \(C^ \infty(E)\), the algebra of all real-valued \(C^ \infty\)-functions on \(E\). The main result is the following Theorem: Let \(\mathcal A\) be a stable and sequentially evaluating algebra type. The class of weakly \({\mathcal A}\)-countably separated spaces is closed under formation of arbitrary products and subspaces. Further, if \((F_ \iota)_{\iota\in I}\) is a family of weakly \({\mathcal A}\)-countably separated spaces such that \(F_ \iota= \text{Hom }{\mathcal A}(F_ \iota)\), then \(E= \text{Hom } {\mathcal A}(E)\) for every weakly closed subspaces of \(F= \prod_{\iota\in I} F_ \iota\). If \(E\) is an complete infra-Schwartz space, i.e. \(E\) is isomorphic to subspaces of products of reflexive Banach spaces, the authors prove that \(E=\text{Hom } C^ \infty(E)\). The authors make use of the so-called sequentially evaluating property of the algebra \(C^ \infty(E)\).
algebra of all real-valued \(C^ \infty\)-functions
stable and sequentially evaluating algebra type
scientific article; zbMATH DE number 572587