Local theory of integral Banach mapping spaces
type: Item
instance of: scholarly article
title: Local theory of integral Banach mapping spaces
zbMATH Open document ID: 1405.46015
author: Pingchuan Dong
author: Haiyi Jiang
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-07-04T00:00:00Z
review text: \textit{D. W. Dean} [Proc. Am. Math. Soc. 40, 146--148 (1973; Zbl 0263.46014)] showed that, if \(E\) is a Banach space, then, for any finite-dimensional Banach space \(L\), the equation \(B(L,E)^{**} = B(L,E^{**})\) holds, where \(B(L,E)\) denotes the bounded linear operators from \(L\) to \(E\). He then used this to prove the principle of local reflexivity. This equation in turn served as inspiration for the definition of local reflexivity for operator spaces. Taking inspiration from the theory of operator spaces, the authors study variants of local reflexivity, exactness and injectivity defined by the operator ideal \({\mathcal I}(E,F)\) of integral operators, from a Banach space \(E\) to a Banach space \(F\), with integral norm. In the paper, the term ``Banach mapping space'' is used instead of the more common ``operator ideal'' or ``Banach operator ideal''. A Banach space \(E\) is integral injective if, for any Banach spaces \(F_0 \subseteq F\), every \(\phi_0 \in {\mathcal I}(F_0,E)\) has a linear extension \(\phi \in {\mathcal I}(F,E)\) with the same integral norm. The authors show that every dual Banach space is integral injective. Curiously, the proof of Theorem~2.3 uses a lemma stated for an operator space \(E\), but the proof of Theorem~2.3 should be clear by the duality of injective tensor products, the fact that injective tensor products respect subspaces, and the Hahn-Banach theorem. The authors say that a Banach space is integral locally reflexive if, for any finite-dimensional Banach space \(L\), we have \({\mathcal I}(L,E^{**}) = {\mathcal I}(L,E)^{**}\). It is shown that any Banach space is integral locally reflexive. This is because finite-dimensional Banach spaces are reflexive and have the metric approximation property so that \[ {\mathcal I}(L,E^{**}) = (L \hat{\otimes}_\varepsilon E^*)^* = (L^* \hat{\otimes}_\pi E)^{**} = {\mathcal I}(L,E)^{**}. \] (Here \(\varepsilon\) and \(\pi\), respectively, denote the injective and projective tensor norms…
reviewed by: Vegard Lima
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 47L20
zbMATH DE Number: 6898331
zbMATH Keywords: integral Banach mapping space
zbMATH Keywords: integral injectivity
zbMATH Keywords: integral local reflexivity
zbMATH Keywords: integral exactness
zbMATH Keywords: integral nuclearity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.05.030
OpenAlex ID: W2804578054
cites work: The Equation L(E, X ∗ ∗ = L(E, X) ∗ ∗ and the Principle of Local Reflexivity
cites work: Q4023934
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cites work: Produits tensoriels topologiques et espaces nucléaires
cites work: The extension property of complex Banach spaces
cites work: Banach Spaces with the Extension Property
cites work: A Theorem of the Hahn-Banach Type for Linear Transformations
cites work: Q4787522
cites work: Q4431278
DOI: 10.1016/J.JMAA.2018.05.030
Wikidata QID: Q129892204
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description: scientific article; zbMATH DE number 6898331