Linear extension operators of bounded norms
type: Item
instance of: scholarly article
title: Linear extension operators of bounded norms
zbMATH Open document ID: 1395.46022
author: Takamitsu Yamauchi
author: Dmitri Shakhmatov
author: V. Valov
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-06-28T00:00:00Z
review text: Recall that a compact Hausdorff space \(X\) is said to be a Dugundji space if, for any embedding \(e\) of \(X\) into another compact Hausdorff space \(Y\), there exists a regular extension operator \(u: C(X) \to C(Y)\), that is, \(u(f) \circ e= f\), for every \(f \in C(X)\), \(\|u\|=1\) and \(u(1_{X})=1_{Y}\), where \(1_{X}\) and \(1_{Y}\) are the constant functions on \(X\) and \(Y\), respectively, taking the value 1, see [\textit{A. Pełczyński}, Diss. Math. 58, 92 p. (1968; Zbl 0165.14603)]. The authors consider the following natural question. Is it true that a space \(X\) is a Dugundji space if there exist an embedding of \(X\) into a Tychonoff space \([0,1]^A\) and a linear extension operator \(u:C(X) \to C([0,1]^A)\) satisfying \(\|u\|<2\)? First, they give a positive answer to this question, when for any \(y \in Y\) with \(u(1_X)(y)=1\) and \(f, g \in C(X)\), we have \(|u(fg)(y)| \leq \|g\| \;|u(|f|)(y)|\). Then, they present some consequences of this result. Other interesting problems related to Dugundji spaces are left open.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 6897104
zbMATH Keywords: Dugundji space
zbMATH Keywords: function space
zbMATH Keywords: extension operator
reviewed by: Elói M. Galego
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.06.030
OpenAlex ID: W2809123282
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Wikidata QID: Q129650553
DOI: 10.1016/J.JMAA.2018.06.030
description: scientific article; zbMATH DE number 6897104