Ideals of operators and the metric approximation property
type: Item
instance of: scholarly article
title: Ideals of operators and the metric approximation property
zbMATH Open document ID: 1068.46014
author: Vegard Lima
author: Åsvald Lima
published in: Journal of Functional Analysis
publication date: 2004-08-16T00:00:00Z
review text: A Banach space \(X\) is said to have the metric approximation property (M.A.P. for short) if its identity operator can be uniformly approximated on compact subsets of \(X\) by finite-rank operators whose norms are \(\leq 1\). According to \textit{J. Johnson} [J. Funct. Anal. 32, 304--311 (1979; Zbl 0412.47024)], if \(X\) has the M.A.P., then for all Banach spaces \(Y\), the finite-rank operators \(F(Y,X)\) form an ideal in the Banach space \(L(Y,X)\) of bounded linear operators from \(Y\) to \(X\). This means that there exists a norm one projection on \((L(Y,X))^\ast\) whose kernel equals the annihilator of \(F(Y,X)\). According to \textit{Å. Lima} [Isr. J. Math. 84, No. 3, 451--475 (1993; Zbl 0814.46016)], if \(X\) has the Radon-Nikodým property and \(F(X,X)\) is an ideal in \(L(X,X)\), then \(X\) has the M.A.P. It is not known whether this result holds for arbitrary Banach spaces \(X\). The main result of this paper asserts that a Banach space \(X\) has the M.A.P. whenever \(F(\widehat{X},X)\) is an ideal in \(L(\widehat{X},X)\) for all equivalent renormings \(\widehat{X}\) of \(X\). Interchanging the roles of \(\widehat{X}\) and \(X\) yields the M.A.P. (with conjugate operators) for \(X^\ast\). Similar results hold for the metric compact approximation property. The proofs rely on results from [\textit{Å. Lima, O. Nygaard} and \textit{E. Oja}, Isr. J. Math. 119, 325--348 (2000; Zbl 0983.46024)] and [\textit{Å. Lima} and \textit{E. Oja}, J. Aust. Math. Soc. 77, No. 1, 91--110 (2004; Zbl 1082.46016)]. They develop further the method of [\textit{Å. Lima} and \textit{E. Oja}, Proc. Am. Math. Soc. 130, No. 12, 3631--3640 (2002; Zbl 1006.46004)] of constructing ``good'' Hahn-Banach extension operators. This can be done thanks to the existence of equivalent locally uniformly rotund renormings that are arbitrarily close to the initial norm (see [\textit{R. Deville, G. Godefroy} and \textit{V. Zizler}, Smoothness and renormings in Banach spaces (Pitman Monographs and Surve…
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 2091742
zbMATH Keywords: metric approximation property
zbMATH Keywords: spaces of operators
zbMATH Keywords: ideals in Banach spaces
reviewed by: Eve Oja
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.10.001
OpenAlex ID: W2085619415
cites work: Self-induced compactness in Banach spaces
cites work: Q3975352
cites work: Q3134552
cites work: Q4319990
cites work: \(M\)-ideals in Banach spaces and Banach algebras
cites work: Remarks on Banach spaces of compact operators
cites work: A complementably universal conjugate Banach space and its relation to the approximation problem
cites work: The metric approximation property, norm-one projections and intersection properties of balls
cites work: Property (wM*) and the unconditional metric compact approximation property
cites work: Isometric factorization of weakly compact operators and the approximation property
cites work: Ideals of compact operators
cites work: Hahn-Banach extension operators and spaces of operators
cites work: An Introduction to Banach Space Theory
cites work: Geometry of Banach spaces having shrinking approximations of the identity
cites work: An Approximation Property Related to M-Ideals of Compact Operators
cites work: Q4183861
cites work: Linear Hahn–Banach extension operators
cites work: The Compact Approximation Property does not imply the Approximation Property
DOI: 10.1016/J.JFA.2003.10.001
description: scientific article; zbMATH DE number 2091742