Amenability and virtual diagonals for von Neumann algebras
type: Item
instance of: scholarly article
title: Amenability and virtual diagonals for von Neumann algebras
zbMATH Open document ID: 0655.46053
DOI: 10.1016/0022-1236(88)90136-X
author: Edward G. Effros
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Bimodules are defined for pairs of unital Banach algebras, and the module elements form a Banach space. If this space is a dual Banach space \(V^*\) then the module is called a normal module when the algebras are von Neumann algebras, and the module actions are assumed \(\sigma\)-weak, weak* continuous. A unital Banach algebra A is said to be amenable if, for any dual A bimodule \(V^*\), any derivation \(\delta:A\to V^*\) is inner. A von Neumann algebra R is amenable if this condition is satisfied for all normal R-bimodules and derivations \(R\to V^*\). A normal virtual diagonal for R is an element \(M\in R{\hat \otimes}^{\sigma}R\) such that \(rM=Mr\), \(\forall r\in R\), and \(\pi (M)=1\) where \(\pi\) is the restriction to \(R\otimes R\) of a certain mapping \(\pi^*_ 0:R{\hat \otimes}_ 0^{\sigma}R\to R\) associated to a certain tensor product construction based on an assumed Grothendieck inequality. It is proved that a von Neumann algebra is amenable iff it has a normal virtual diagonal, and, as a corollary, that all nuclear \(C^*\)-algebras are amenable. The corollary was proved earlier by Haagerup. But the present paper yields a direct proof of Haagerup's result, the basic idea being that a certain universal bimodule over a given von Neumann algebra is automatically normal.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 4069599
zbMATH Keywords: Bimodules
zbMATH Keywords: pairs of unital Banach algebras
zbMATH Keywords: dual Banach space
zbMATH Keywords: normal module
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: module actions are assumed \(\sigma\)-weak, weak* continuous
zbMATH Keywords: derivations
zbMATH Keywords: tensor product construction based on an assumed Grothendieck inequality
zbMATH Keywords: normal virtual diagonal
zbMATH Keywords: all nuclear \(C^*\)-algebras are amenable
zbMATH Keywords: universal bimodule
reviewed by: Palle Jorgensen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4069599