(Non-)Dunford-Pettis operators on noncommutative symmetric spaces
type: Item
instance of: scholarly article
title: (Non-)Dunford-Pettis operators on noncommutative symmetric spaces
zbMATH Open document ID: 1491.46059
author: Jinghao Huang
author: M. Pliev
author: Fedor Sukochev
published in: Journal of Functional Analysis
publication date: 2022-03-22T00:00:00Z
review text: An operator \(T\) from a Banach space \(X \) into another Banach space \(Y\) is called a Dunford-Pettis operator (shortly, (DP)-operator), whenever $T$ maps weakly null sequences into norm null sequences. A Banach space $X$ has the Dunford-Pettis property (shortly, DPP) if all weakly compact operators defined from $X$ into an arbitrary Banach space $Y$ are Dunford-Pettis operators, or equivalently, if for all weakly null sequences \((x_n)\) in \(X\) and \((f_n)\) in \(X^*\), we have \(f_n(x_n) \to 0\). The authors consider two important questions concerned with a von Neumann algebra. One asks: if an operator on the predual \(\mathcal M_*\) of a von Neumann algebra \(\mathcal M\) acting on a separable Hilbert space \(\mathcal H\) is a non-Dunford-Pettis operator, is there a subspace \(Z\) of \(\mathcal M_*\) such that \(Z\) is isomorphic to \(\bigoplus_{\ell_1}\ell_2\) with \(T|_Z\) an isomorphism? They give the following characterizations: Theorem 1. Let \(\mathcal M\) be a von Neumann algebra acting on a separable Hilbert space \(\mathcal H\); then the following statements are equivalent: \((1)\) \(\mathcal M=\bigoplus_{1\leq n<\infty}(\mathcal A_n\otimes B(\ell_2^n))\), where the\(\mathcal A_n\)'s are commutative von Neumann algebras and at least one of them is not atomic; \((2)\) \(\mathcal M_*\) is isomorphic to one of the following spaces \[ L_1(0,1),\ S_1\oplus L_1(0,1), \ L_1(S_1),\] where \(S_1\) is the \(\ell_1\)-sum of the preduals to \(B(\ell_2^n)\), \(n=1,2,\dots\); \((3)\) any non-(DP)-operator \(T\) on \(\mathcal M_*\) fixes a copy of \(\oplus_{\ell_1}(\ell_2)\). Another problem is: for which symmetric function spaces \(E(0,\infty)\) and semifinite von Neumann algebras \(\mathcal M\) (equipped with semifinite faithful normal traces \(\tau\)) do the noncommutative symmetric spaces \(E(\mathcal M, \tau)\) have the DPP? They answer this problem and fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann alge…
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46B85
zbMATH DE Number: 7495781
zbMATH Keywords: (non-)Dunford-Pettis operator
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: predual of a von Neumann algebra
zbMATH Keywords: noncommutative symmetric space
reviewed by: Guoxing Ji
MaRDI profile type: Publication
full work available at URL: j.jfa.2022.109443
OpenAlex ID: W4214880466
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DOI: 10.1016/J.JFA.2022.109443
description: scientific article; zbMATH DE number 7495781