A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure
type: Item
instance of: scholarly article
title: A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure
zbMATH Open document ID: 1123.46044
author: P. G. Dodds
author: O. Ye. Tikhonov
author: Theresa K.-Y. Dodds
author: Fedor Sukochev
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: Let \(M\) be a semifinite von Neumann algebra with a normal faithful semifinite trace \(\tau\). Let \(E(M,\tau)\) be a noncommutative symmetric space associated with \((M,\tau)\) corresponding to a symmetric function space \(E\). Using the noncommutative Köthe duality, the following general form of the Yosida-Hewitt decomposition is proved: Suppose that \(M\subset E(M,\tau)\) or that \(E\) is fully symmetric. If \(\varphi\in E(M,\tau)^\ast\), then there exists a unique decomposition \(\varphi=\tau(a\cdot)+\varphi_s\,,\) where \( a\) is from the Köthe dual of \(E(M,\tau)\) and \(\varphi_s\in E(M,\tau)^\ast\) is singular. (A functional is singular if it vanishes on some order dense ideal.) This permits to define the Yosida-Hewitt projection and characterize, in terms of this projection, bounded convex sets in \(E(M,\tau)\) which are closed locally in measure. Based on this, a Hahn-Banach type result as well as the finite intersection property for convex sets closed locally in measure are established.
reviewed by: Jan Hamhalter
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5070771
zbMATH Keywords: non-commutative Banach function spaces
zbMATH Keywords: singular functionals
zbMATH Keywords: measurable operators
zbMATH Keywords: local convergence in measure
zbMATH Keywords: Köthe duality
MaRDI profile type: Publication
full work available at URL: s11117-005-1384-0
OpenAlex ID: W2044254106
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DOI: 10.1007/S11117-005-1384-0
description: scientific article; zbMATH DE number 5070771