Norm-linear and norm-additive operators between uniform algebras
type: Item
instance of: scholarly article
title: Norm-linear and norm-additive operators between uniform algebras
zbMATH Open document ID: 1171.47032
author: Thomas V. Tonev
author: Rebekah B. Johnson Yates
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-23T00:00:00Z
review text: Let \(A\subseteq C(X)\) and \(B\subseteq C(Y)\) be uniform algebras with Choquet boundaries \(\delta A\) and \(\delta B\), respectively, and let \({\mathbb R}_+\) denote the set of nonnegative reals numbers. The authors call a map \(T:A\to B\) norm-additive in modulus if \[ \big\| |Tf|+|Tg|\big\|=\big\| |f|+|g|\big\|\qquad (f,g\in A). \] It is then shown that an \({\mathbb R}_+\)-homogeneous, norm-additive in modulus surjection \(T:A\to B\) is induced on the Choquet boundary by a continuous map \(\tau:\delta A\to\delta B\) in the sense that \(|(Tf)(\tau(x))|=|f(x)|\) for every \(x\in \delta A\) and every \(f\in A\). Actually, \(\tau\) is a homeomorphism whenever \(T\) is bijective. The key idea is to show that, for each \(x\in\delta A\), the set \[ E_x:=\bigcap_{f\in A\atop |f(x)|=\|f\|} |T(f)|^{-1}\{\|Tf\|\} \] is a singleton and belongs to \(\delta B\). Several applications follow where norm conditions on a surjection \(T:A\to B\), which force it to be an algebra isomorphism, are studied. The list is too long to be given here, but as a sample example, we note that it is shown that if a surjection \(T\) between uniform algebras satisfies (i) \(\|Tf+\alpha Tg\|=\|f+\alpha g\|\), \(|\alpha|=1\), and (ii) \(T\) is unital and \(T(\sqrt{-1})=\sqrt{-1}\), then \(T\) is an isometric algebra isomorphism. The same conclusion holds if (ii) is replaced by (ii') \(T\) preserves the peripheral spectrum of \(\mathbb C\)-peaking functions from \(A\).
reviewed by: Bojan Kuzma
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 5568968
zbMATH Keywords: uniform algebra
zbMATH Keywords: peaking function
zbMATH Keywords: peak set
zbMATH Keywords: generalized peak point
zbMATH Keywords: Choquet boundary
zbMATH Keywords: Shilov boundary
zbMATH Keywords: peripheral spectrum
zbMATH Keywords: homeomorphism
zbMATH Keywords: algebra isomorphism
zbMATH Keywords: norm-additivity
zbMATH Keywords: norm-linearity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.039
OpenAlex ID: W2050512072
cites work: Q5481265
cites work: Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties
cites work: Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative
cites work: Continuous mappings induced by isometries of spaces of continuous functions
cites work: Q4252796
cites work: A characterization of multiplicative linear functionals in Banach algebras
cites work: Uniform algebra isomorphisms and peripheral multiplicativity
cites work: Algebra isomorphisms between standard operator algebras
cites work: Q5447325
cites work: Q5624218
cites work: Selected preserver problems on algebraic structures of linear operators and on function spaces
cites work: Some characterizations of the automorphisms of \(B(H)\) and \(C(X)\)
cites work: Multiplicatively spectrum-preserving maps of function algebras
cites work: MULTIPLICATIVELY SPECTRUM-PRESERVING MAPS OF FUNCTION ALGEBRAS. II
cites work: Q5755151
cites work: Q5612756
DOI: 10.1016/J.JMAA.2009.03.039
description: scientific article; zbMATH DE number 5568968