New criterion of the approximation property
type: Item
instance of: scholarly article
title: New criterion of the approximation property
zbMATH Open document ID: 1154.46008
author: Ju Myung Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-07-10T00:00:00Z
review text: For Banach spaces \(X\) and \(Y\), let the Banach space of all bounded linear operators from \(X\) to \(Y\) be denoted by \({\mathcal L}(X,Y)\). Let \({\mathcal F}(X,Y)\) and \({\mathcal K}(X,Y)\) denote its subspaces of finite-rank and compact operators. A Banach space \(X\) is said to have the approximation property (AP) if the identity map on \(X\) belongs to the closure of \({\mathcal F}(X,X)\) in the topology \(\tau\) of uniform convergence on compact subsets of \(X\). By a well-known result of Grothendieck, \(X\) has the AP if and only if \({\mathcal F}(Y,X)\) is \(\| \cdot\| \)-dense in \({\mathcal K}(Y,X)\) for every Banach space \(Y\). The author proves that \(X\) has the AP if and only if \({\mathcal F}(X,Y)\) is \(\tau\)-dense in \({\mathcal K}(X,Y)\) for every Banach space \(Y\) (or just for every separable reflexive Banach space \(Y\)). Reviewer's remark. This gives a new proof to Grothendieck's result that the AP of \(X^\ast\) implies the AP of \(X\). Indeed, if \(X^\ast\) has AP, then \({\mathcal F}(X,Y)\) is \(\| \cdot\| \)-dense in \({\mathcal K}(X,Y)\) for all \(Y\) and hence also \(\tau\)-dense, which means that \(X\) has the AP. The traditional proof relies on the trace condition (``condition de biunivocité'') of Grothendieck.
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 5299492
zbMATH Keywords: approximation property
zbMATH Keywords: topology of compact convergence
zbMATH Keywords: compact operators
reviewed by: Eve Oja
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.05.002
OpenAlex ID: W2079493371
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DOI: 10.1016/J.JMAA.2008.05.002
description: scientific article; zbMATH DE number 5299492