An optimal linear approximation for a class of nonlinear operators between uniform algebras
type: Item
instance of: scholarly article
title: An optimal linear approximation for a class of nonlinear operators between uniform algebras
zbMATH Open document ID: 1400.46020
author: Pei-Kee Lin
author: Yunbai Dong
author: Bentuo Zheng
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-08-15T00:00:00Z
review text: Let \(X,Y\) be compact Hausdorff spaces and let \(A,B\) be uniform algebras on \(X,Y\), respectively, and let \(\partial A, \partial B\) be the respective Šilov boundaries. This interesting paper deals with the structure of maps \(T: A \rightarrow B\) such that \(T(1) = 1\), \(T\) maps the set of invertible elements of \(A\) onto the set of invertible elements in \(B\) and satisfies certain norm inequalities. Results of this nature for spectrally multiplicative maps when the uniform algebra is considerd on its maximal ideal spaces were obtained by \textit{L. Molnár} [Proc. Am. Math. Soc. 130, No. 1, 111--120 (2002; Zbl 0983.47024)], \textit{N. V. Rao} and \textit{A. K. Roy} [Proc. Edinb. Math. Soc., II. Ser. 48, No. 1, 219--229 (2005; Zbl 1074.46033)]. The authors show that the conditions on \(T\) imply that the Šilov boundaries are homeomorphic by a homeomorphism \(\tau\) and obtain pointwise inequalities on the Šilov boundary comparing \(T(f)\) and \(f \circ \tau\). An interesting step in the proof is to get pointwise estimates for maps defined only on the set of invertible elements that preserve constants (Theorem 2.4).
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 6918459
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: uniform algebras
zbMATH Keywords: Šilov boundary
zbMATH Keywords: composition operators
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.07.016
OpenAlex ID: W2884587121
cites work: On isomorphisms of continuous function spaces
cites work: Linear isometries between subspaces of continuous functions
cites work: The representations of linear functionals by measures on sets of extreme points
cites work: Q5597848
cites work: A Generalized Banach-Stone Theorem
cites work: On Isomorphisms with Small Bound
cites work: Isomorphisms of \(C_ 0(Y)\) onto \(C(X)\)
cites work: Coarse version of the Banach-Stone theorem
cites work: A Bound-Two Isomorphism Between C(X) Banach Spaces
cites work: Perturbations of isometries between C(K)-spaces
cites work: An optimal nonlinear extension of Banach-Stone theorem
cites work: Q4801656
cites work: Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties
cites work: Continuous mappings induced by isometries of spaces of continuous functions
cites work: Q5609643
cites work: Nonlinear generalizations of the Banach-Stone theorem
cites work: Q5447325
cites work: Uniform algebra isomorphisms and peripheral multiplicativity
cites work: Some characterizations of the automorphisms of \(B(H)\) and \(C(X)\)
cites work: Q5508589
cites work: Multiplicatively spectrum-preserving maps of function algebras
cites work: MULTIPLICATIVELY SPECTRUM-PRESERVING MAPS OF FUNCTION ALGEBRAS. II
cites work: Q3272732
cites work: Applications of the Theory of Boolean Rings to General Topology
cites work: Spectral conditions for almost composition operators between algebras of functions
DOI: 10.1016/J.JMAA.2018.07.016
description: scientific article; zbMATH DE number 6918459