On positive embeddings of \(C(K)\) spaces into \(C(S,X)\) lattices
type: Item
instance of: scholarly article
title: On positive embeddings of \(C(K)\) spaces into \(C(S,X)\) lattices
zbMATH Open document ID: 1471.47020
author: Michael A. Rincón-Villamizar
author: Elói M. Galego
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-08-27T00:00:00Z
review text: A result of \textit{G. Plebanek} [Stud. Math. 216, No. 2, 179--192 (2013; Zbl 1286.46026)] established that a positive isomorphism from $C(K)$ into (not necessarily onto) $C(S)$ for compact \(K\) and \(S\) implies the existence of a~map \(\psi\) from $S$ into the collection of all subsets of \(K\) with cardinally less than or equal to a~number \(n\) which is upper semicontinuous and \(\cup \{\psi (s): s \in S\}=K\). Here, the authors extend this result to maps from \(C(K)\) into \(C(S,X)\) for a~class of Banach-lattices. These Banach-lattices \(X\) have the property that \(\lambda ^+(X) = \operatorname{inf}\{\max ||x-y||,||x+y|| : x \text{ and }y \text{ are both positive with norm }1\}>1\). A~variety of results and examples are discussed in relation to values of \(\lambda^+\). Furthermore, it is shown that, for \(K\) and \(S\) locally compact and \(X\) a Banach lattice with \(\lambda^+ (X) >1\), if a positive isomorphism \(T:C_0(K)\rightarrow C_0(S,X)\) (where the subscript \(0\) denotes functions vanishing at infinity) satisfies \(\|T\| \,\|T^{-1}\|< \lambda ^+(X)\), then for all ordinals \(\alpha\), \(K^{(\alpha)}\) is a continuous image of a subspace of \(S^{(\alpha)}\). The set \(K ^{(\alpha)}\) (or \(S^{(\alpha)}\)) is defined recursively by deleting isolated points. The paper includes several unsolved problems suggested by these results.
reviewed by: William A. Feldman
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 00A27
zbMATH DE Number: 6925413
zbMATH Keywords: positive isomorphism
zbMATH Keywords: Banach lattice valued functions vanishing at infinity
Wikidata QID: Q111288310
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.08.003
OpenAlex ID: W2886324571
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DOI: 10.1016/J.JMAA.2018.08.003
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description: scientific article; zbMATH DE number 6925413