Arens algebras, associated with commutative von Neumann algebras
type: Item
instance of: scholarly article
title: Arens algebras, associated with commutative von Neumann algebras
zbMATH Open document ID: 0924.46049
published in: Annales Mathématiques Blaise Pascal
publication date: 1999-11-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79195
review text: The classification of the Arens algebras associated with the commutative von Neumann algebras is given. Let \(M\) be a von Neumann algebra with a faithful normal finite trace \(\mu\). The \(L^p(X,\mu)\) space \((1\leq p<\infty)\) is defined as the space of all \(\mu\)-measurable operators affiliated with \(M\) for which \(\mu((x^\ast x)^p)<\infty\). The Arens algebra \(L^\omega(M, \mu)\) is then defined as an intersection \[ L^\omega(M,\mu)=\bigcap_{1\leq p<\infty}L^p(M,\mu). \] It is shown that \(L^\omega(M,\nu)\) and \(L^\omega(M,\mu)\) are \(\ast\)-isomorphic whenever \(M\) and \(N\) are non-atomic commutative von Neumann algebras acting on a separable Hilbert space with faithful normal finite traces \(\mu\) and \(\nu\), respectively. The main result of the paper fully describes the situation when \(L^\omega(M,\mu)\) and \(L^\omega(M,\nu)\) are \(\ast\)-isomorphic for a non-atomic commutative \(\sigma\)-finite von Neumann algebra \(M\) in terms of the decomposition of its projection Boolean algebra into a direct sum of homogeneous parts. Interesting consequences of this result are given.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1187661
zbMATH Keywords: Arens algebras
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: faithful normal finite trace
zbMATH Keywords: space of all \(\mu\)-measurable operators
author: R. Z. Abdullaev
author: Vladimir Chilin
reviewed by: Jan Hamhalter
MaRDI profile type: Publication
OpenAlex ID: W2319955986
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DOI: 10.5802/AMBP.101
description: scientific article; zbMATH DE number 1187661