On peak sets for certain function spaces
type: Item
instance of: scholarly article
title: On peak sets for certain function spaces
zbMATH Open document ID: 0679.46038
author: Shinya Yamaguchi
author: Junzo Wada
published in: Tokyo Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The authors consider some generalizations to function spaces, of results on function algebras. Let \(X\) be a compact Hausdorff space and let \(C(X)\) denote the Banach space of complex-valued continuous function on \(X\). A function space on \(X\) is a closed subspace of \(C(X)\) containing the constant function and separating the points of \(X\). For a function space \(A\) on \(X\) let \(A(X)=\{f\in C(X): fa\in A\) for all \(a\in A\}\). The authors consider spaces satisfying the following condition: (A) Any peak set for \(A\) is a peak set for \(A(X)\). Any function algebra on \(X\) satisfied of course condition (A); the authors give several examples of function spaces (which are not algebras) which satisfy condition (A). Among results obtained are the following: Suppose \(A\) satisfies condition (A). If Every peak set for \(\Re A\) is a peak set for \(A\) then \(A=C(X)\). Also, if \(X\) is the union of countably many interpolation sets for \(A\), then \(A=C(X)\).
reviewed by: E. Briem
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 4113299
zbMATH Keywords: function spaces
zbMATH Keywords: function algebras
zbMATH Keywords: peak set
zbMATH Keywords: interpolation sets
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270133986
OpenAlex ID: W1964774816
DOI: 10.3836/TJM/1270133986
description: scientific article; zbMATH DE number 4113299