Local 2-cocycles.
type: Item
instance of: scholarly article
title: Local 2-cocycles.
zbMATH Open document ID: 1055.47031
published in: Applied Mathematics. Series B (English Edition)
publication date: 2002-01-01T00:00:00Z
review text: Let \(R\) be a von Neumann algebra and \(S\) be a unital dual Banach \(R\)-bimodule. A bounded bilinear mapping \(\Phi\colon R\times R\to S\) is called a local \(2\)-cocycle if, for each pair \((a,b)\in R\times R\), there exists a bounded bilinear mapping \(\Phi_{a,b}\colon R\times R\to S\) such that \(\Phi(a,b)=\Phi_{a,b}(a,b)\) and \(\Phi_{a,b}\) satisfies the cocycle identity \[ x\Phi_{a,b}(y,z)-\Phi_{a,b}(xy,z)+\Phi_{a,b}(x,yz)-\Phi_{a,b}(x,y)z=0 \qquad(x,y,z\in R).\tag{1} \] The author shows that every local \(2\)-cocycle is in fact a global \(2\)-cocycle, that is, it satisfies identity~(1). The proof proceeds in tedious but straightforward computations.
reviewed by: Martin Mathieu
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 1836079
zbMATH Keywords: \(2\)-cocycles
zbMATH Keywords: von Neumann algebras
author: Jianhua Zhang
MaRDI profile type: Publication
full work available at URL: s11766-002-0006-x
OpenAlex ID: W195815325
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DOI: 10.1007/S11766-002-0006-X
description: scientific article; zbMATH DE number 1836079