Characterization of the spectra of certain function algebras
type: Item
instance of: scholarly article
title: Characterization of the spectra of certain function algebras
zbMATH Open document ID: 0805.46051
DOI: 10.1007/BF01199105
author: Peter Biström
author: Mikael Lindström
published in: Archiv der Mathematik
publication date: 1995-02-09T00:00:00Z
review text: 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 \(C^ \infty(E)\).
reviewed by: Rudolf Rupp
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 572587
zbMATH Keywords: spectrum
zbMATH Keywords: algebra of all real-valued \(C^ \infty\)-functions
zbMATH Keywords: stable and sequentially evaluating algebra type
zbMATH Keywords: complete infra-Schwartz space
MaRDI profile type: Publication
cites work: Q3990748
cites work: Homomorphisms on some function algebras
cites work: On compactness in locally convex spaces
cites work: Q3918541
cites work: More Smoothly Real Compact Spaces
cites work: Q3709279
full work available at URL: https://doi.org/10.1007/bf01199105
OpenAlex ID: W2351998447
description: scientific article; zbMATH DE number 572587