Isometries of non-commutative Lorentz spaces
type: Item
instance of: scholarly article
title: Isometries of non-commutative Lorentz spaces
zbMATH Open document ID: 0646.46060
DOI: 10.1007/BF01160954
author: Vladimir Chilin
author: Arthur M. Medzhitov
author: Fedor Sukochev
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174024
review text: Let \(\mathfrak A\) be a finite von Neumann algebra, \(\mu\) be a faithful normal finite trace on \(A\), \(L_1(\mathfrak A,\mu)\) be the Banach space of all \(\mu\)-integrable measurable operators affiliated with \(\mathfrak A\), \(E\) be a subspace in \(L_1(\mathfrak A,\mu)\), \(\| \cdot \| _E\) be a norm on \(E\). The pair \((E,\| \cdot \| _E)\) is said to be a symmetric (respectively a fully symmetric) space on (\(\mathfrak A,\mu)\) if from \(x\in E\), \(y\in L_1(\mathfrak A,\mu)\) and \(\tilde y\le \tilde x\) (respectively \(\int^s_0 \tilde y(t)\,dt\le \int^s_0\tilde x(t)\,dt\), \(s\in (0,\mu(\mathbf{1}))\) it follows that \(y\in E\) and \(\| y\| _E\le \| x\| _E\), where \(\tilde x(t)\) denotes the rearrangement of \(x\). In the paper the general form of a positive linear isometry of a symmetric space onto a fully symmetric space is determined. Also is given a full characterization of all linear isometries of non-commutative Lorentz spaces.
reviewed by: Vladimir Chilin
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 4054345
zbMATH Keywords: finite von Neumann algebra
zbMATH Keywords: faithful normal finite trace
zbMATH Keywords: rearrangement
zbMATH Keywords: positive linear isometry of a symmetric space onto a fully symmetric space
zbMATH Keywords: characterization of all linear isometries of non-commutative Lorentz spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4054345