Automatic continuity of biseparating homomorphisms defined between groups of continuous functions
type: Item
instance of: scholarly article
title: Automatic continuity of biseparating homomorphisms defined between groups of continuous functions
zbMATH Open document ID: 1206.22003
author: Salvador Hernández
author: Ana María Ródenas
author: M. V. Ferrer
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
review text: The Banach-Stone theorem associates to every linear isometry \(T: C(X,\mathbb R)\to C(Y,\mathbb R)\), \(X\) and \( Y\) compact Hausdorff spaces, a homeomorphism \(\phi: Y\to X\) in such a way that \(Tf= (T1) \cdot (f\circ \phi)\) for every \(f \in C(X,\mathbb R)\). It follows that the Banach space structure (linear and metric structure) of \(C(X,\mathbb R)\) completely determines the topological space \(X\). The metric structure here can be replaced by some additional algebraic structure. One of the most efficient ways to do this is by so-called separating maps. A linear operator \(T\) as above is said to be separating provided \(Tf\cdot Tg=0\) when \(f\cdot g=0\) and a linear bijection \(T\) is said to be biseparating provided both \(T\) and \(T^{-1}\) are separating. The present paper addresses similar questions for the group \(C(X,\mathbb T)\) of continuous functions of \(X\) into the unit circle \(\mathbb T\) of the complex plane. In order to investigate which group isomorphisms \(H: C(X,\mathbb T)\to C(Y,\mathbb T)\) are associated to homeomorphisms between \(X\) and \(Y\), the authors say that a map \(H: C(X,\mathbb T)\to C(Y,\mathbb T)\) is separating if whenever \(f,g \in C(X,\mathbb T)\) are such that \((f(x)-1)(g(x)-1)=0\) for every \(x\in X\), then \((Hf(y)-1)(Hg(y)-1)=0\) for every \(y\in Y\). This concept, that adds a pinch of the additive structure of \(C(X,\mathbb C)\) in the multiplicative group \(C(X,\mathbb T)\), is shown to be a good analog in \(C(X,\mathbb T)\) of the concept of separating map. Automatic continuity theorems and representations as composition operators that are typical of the linear theory (see, for instance, \textit{S. Hernández, E. Beckenstein} and \textit{L. Narici} [Manuscr. Math. 86, 409--416 (1995; Zbl 0827.46032)] or \textit{J. Araujo, E. Beckenstein} and \textit{L. Narici} [J. Math. Anal. Appl. 192, 258--265 (1995; Zbl 0828.47024)], the paper's list of references contains many more examples) are proved in this paper for bi…
reviewed by: Jorge Galindo
Mathematics Subject Classification ID: 22A10
Mathematics Subject Classification ID: 22B99
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 5710129
zbMATH Keywords: automatic continuity
zbMATH Keywords: Banach-Stone map
zbMATH Keywords: dual group
zbMATH Keywords: Pontryagin-van Kampen duality
zbMATH Keywords: group-valued continuous function
zbMATH Keywords: group homomorphisms
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.069
OpenAlex ID: W2091166627
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DOI: 10.1016/J.TOPOL.2009.04.069
description: scientific article; zbMATH DE number 5710129