Ideals of operators, approximability in the strong operator topology, and the approximation property
type: Item
instance of: scholarly article
title: Ideals of operators, approximability in the strong operator topology, and the approximation property
zbMATH Open document ID: 1069.46013
published in: Michigan Mathematical Journal
publication date: 2004-11-01T00:00:00Z
review text: The authors give characterisations of the approximation property of a Banach space \(X\) or its dual in terms of pointwise approximability of weakly compact operators by finite rank operators. One of their results says that \(X^*\) has the approximation property if and only if for every Banach space \(Z\) and every weakly compact operator \(T: Z\to X\) there is a net of finite rank operators \(T_\alpha : Z\to X\) of norm \(\| T_\alpha \| \leq \| T\| \) converging to \(T\) pointwise. They also give characterisations of when, for example, for all \(Z\), the space of compact operators \(K(Z,X)\) is an ideal in the space of weakly compact operators \(W(Z,X)\) or \(W(Z,X^{**})\) in the same spirit. (A subspace of a Banach space is called an ideal if its annihilator is the kernel of a norm-\(1\) projection in the dual.) These considerations permit the authors to exhibit a space of compact operators \(K(Z,X)\) that is not an \(M\)-ideal in \(K(Z,X^{**})\) although \(X\) is an \(M\)-ideal in \(X^{**}\). This surprising example solves a problem left open in the reviewer's paper [Math.\ Scand.\ 61, No. 1, 149--164 (1987; Zbl 0619.46062)] where this property of \(K(Z,X)\) is shown to hold if \(X^{**}\) has the approximation property.
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 47L05
zbMATH DE Number: 2112900
zbMATH Keywords: approximation property
zbMATH Keywords: ideals of operators
zbMATH Keywords: weakly compact operators
zbMATH Keywords: \(M\)-ideal
author: Åsvald Lima
author: Eve Oja
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1091112074
OpenAlex ID: W2006374080
cites work: Q4039882
cites work: Spaces of compact operators and their dual spaces
cites work: Q4319990
cites work: Q3684673
cites work: Duality in spaces of operators and smooth norms on Banach spaces
cites work: Produits tensoriels topologiques et espaces nucléaires
cites work: \(M\)-ideals in Banach spaces and Banach algebras
cites work: Separable lifting property and extensions of local reflexivity
cites work: The metric approximation property, norm-one projections and intersection properties of balls
cites work: Isometric factorization of weakly compact operators and the approximation property
cites work: Q4239570
cites work: Hahn-Banach extension operators and spaces of operators
cites work: Ideals of compact operators
cites work: On James's paper 'Separable conjugate spaces'
cites work: Q4136031
cites work: Géométrie des espaces de Banach ayant des approximations de l'identité contractantes
cites work: Geometry of Banach spaces having shrinking approximations of the identity
cites work: The approximation property in terms of the approximability of weak\(^*\)-weak continuous operators.
cites work: Q4172484
cites work: Extremal structure in operator spaces
cites work: M-structure in tensor products of Banach spaces.
DOI: 10.1307/MMJ/1091112074
Recommended article: On the compact approximation property
Recommended article: Q4239570
Recommended article: Isometric factorization of weakly compact operators and the approximation property
Recommended article: Uniqueness of norm-preserving extensions of functionals on the space of compact operators
Recommended article: 𝑀-ideals of compact operators are separably determined
Recommended article: Q3781563
Recommended article: Approximation properties determined by operator ideals and approximability of homogeneous polynomials and holomorphic functions
Recommended article: A remark on the approximation of \(p\)-compact operators by finite-rank operators
Recommended article: Unconditional ideals of finite rank operators
Recommended article: SOME RESULTS OF THE -APPROXIMATION PROPERTY FOR BANACH SPACES
description: scientific article; zbMATH DE number 2112900