Remarks on perturbations of function algebras
type: Item
instance of: scholarly article
title: Remarks on perturbations of function algebras
zbMATH Open document ID: 0721.46031
published in: Tokyo Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let A be a Banach algebra. By an \(\epsilon\)-perturbation of A we mean any multiplication \(\times\) defined on the Banach space A such that \(\| f\times g-f\cdot g\| \leq \epsilon \| f\| \| g\|\), for f,g\(\in A\). An algebra A is said to be \(\epsilon\)-stable if for any \(\epsilon\)-perturbation \(\times\) of A the algebras A and (A,\(\times)\) are isomorphic. The author proves that if \(\{A_{\lambda}\}\) is a family of \(\epsilon\)- stable function algebras then \(\oplus_{\lambda}A_{\lambda}\), a direct sum of algebras \(\{A_{\lambda}\}\), is also \(\epsilon '\)-stable, for some \(\epsilon '>0\). The converse implication is proved under an additional assumption that the Choquet boundary of \(A_{\lambda}\) is connected for each \(\lambda\in \Lambda\). It seems to be an open question whether the converse implication holds without this additional assumption.
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 4188496
zbMATH Keywords: Banach algebra
zbMATH Keywords: \(\epsilon \) -perturbation
zbMATH Keywords: \(\epsilon \) -stable function algebras
zbMATH Keywords: direct sum of algebras
zbMATH Keywords: Choquet boundary
author: Junzo Wada
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270141793
OpenAlex ID: W2012231715
DOI: 10.3836/TJM/1270141793
description: scientific article