Polynomial identification in uniform and operator algebras
type: Item
instance of: scholarly article
title: Polynomial identification in uniform and operator algebras
zbMATH Open document ID: 1220.46032
author: Dillon Ethier
author: Tova Lindberg
author: Aaron Luttman
published in: Annals of Functional Analysis
publication date: 2010-11-23T00:00:00Z
full work available at URL: https://eudml.org/doc/232766
full work available at URL: AFA-tex v1 n1 a11.pdf
review text: The authors establish various criteria for the equality of two elements in some unital Banach algebras. The first part of the paper is devoted to uniform algebras. Let \(\mathcal A\) be a uniform algebra on a compact Hausdorff space \(K\), and let \(f,g \) be two elements of \(\mathcal A\). Among other results, the authors show that, if there exist \(\alpha, \beta \in \mathbb C \setminus \{ 0 \} \) and \(\gamma \in \mathbb C\) such that \( \| \alpha f + \beta h + \gamma \| = \| \alpha g + \beta h + \gamma \|\) for all \(\mathbb C\)-peaking functions \(h \in \mathcal A\), then \(f=g\). Recall that a nonzero element \(h \in \mathcal A\) is called a \(\mathbb C\)-peaking function if its peripheral spectrum is a singleton. In the second part, the authors consider standard operator algebras. Let \(\mathcal A\) be a unital standard operator algebra on a Banach space \(X\) and let \(A, B\) be in \( \mathcal A\). The authors show that, if there exist \(\alpha, \beta \in \mathbb C \setminus \{ 0 \} \) and \(\gamma \in \mathbb C\) such that \( \rho (\alpha A+ \beta T+ \gamma)= \rho (\alpha B+ \beta T+ \gamma)\) for all \(T \in \mathcal A\), where \(\rho (\cdot)\) denotes the spectral radius, then \(A=B\). Other identification criteria are given.
reviewed by: Nadia Boudi
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 47C05
zbMATH DE Number: 5818627
zbMATH Keywords: uniform algebras
zbMATH Keywords: standard operator algebras
zbMATH Keywords: peripheral spectrum
zbMATH Keywords: polynomial identification
zbMATH Keywords: spectral preserver problems
MaRDI profile type: Publication
OpenAlex ID: W1979801888
DOI: 10.15352/AFA/1399900997
description: scientific article; zbMATH DE number 5818627