A polynomially spectrum preserving map between uniform algebras
type: Item
instance of: scholarly article
title: A polynomially spectrum preserving map between uniform algebras
zbMATH Open document ID: 1241.47032
author: Go Hirasawa
author: Hirokazu Oka
published in: Nihonkai Mathematical Journal
publication date: 2011-05-02T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.nihmj/1303477708
review text: Let \(A\) be a uniform algebra on a compact Hausdorff space \(X\) and \(f \in A\). The subset \(\{z \in \sigma(f): |z|= \|f\|\}\) of the spectrum \(\sigma(f)\) is called the \textit{peripheral spectrum} and denoted by \(\sigma_\pi(f)\). The authors characterized surjective maps \(T\) between two uniform algebras \(A\) and \(B\) such that \[ \sigma_\pi(p(T(f), T(g))) \subset \sigma_\pi(p(f, g)) \quad \text{for all} \quad f, g \in A, \] where \(p\) is a polynomial in two variables with complex coefficients of the form \(p(z, w)= zw+ az+ bw+ c\). They show that if \(a \neq b\), then \(T\) is an algebraic isomorphism induced by a homeomorphism from the maximal ideal space \(M_B\) onto \(M_A\); if \(a=b\), then \(T(f)(y)= \eta(y)f(\Phi(y))+ a(\eta(y)-1)\), where \(\eta: M_B \to \{-1, 1\}\) is a continuous map and \(\Phi: M_B \to M_A\) is an onto homeomorphism. Such polynomially spectrum preserving maps were first considered in an unpublished result of O.\,Hatori, T.\,Miura and H.\,Takagi, where the polynomials are more restricted as \(p(z, w)= zw+ az+ bw+ ab\).
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 5883056
zbMATH Keywords: uniform algebras
zbMATH Keywords: isomorphisms
zbMATH Keywords: spectrum preserving maps
reviewed by: Ying-Fen Lin
MaRDI profile type: Publication
cites work: Q5597848
cites work: Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative
cites work: Generalizations of spectrally multiplicative surjections between uniform algebras
cites work: Uniform algebra isomorphisms and peripheral multiplicativity
cites work: Some characterizations of the automorphisms of \(B(H)\) and \(C(X)\)
description: scientific article; zbMATH DE number 5883056