The Haagerup norm on the tensor product of operator modules
type: Item
instance of: scholarly article
title: The Haagerup norm on the tensor product of operator modules
zbMATH Open document ID: 0828.46048
author: Bojan Magajna
published in: Journal of Functional Analysis
publication date: 1995-06-14T00:00:00Z
review text: The author proves that the (analogy of the) Haagerup norm on the tensor product of submodules of \({\mathcal B} ({\mathcal H})\) over a von Neumann algebra \({\mathcal T} \subseteq {\mathcal B} ({\mathcal H})\) is injective. If \({\mathcal R} \subseteq {\mathcal S} \subseteq {\mathcal B} ({\mathcal H})\) are von Neumann algebras with \({\mathcal S}\) injective and \({\mathcal T}= {\mathcal R}' \cap {\mathcal S}\), then the natural map from \({\mathcal S}\otimes_{\mathcal T} {\mathcal S}\) equipped with the Haagerup norm to \(\text{CB} ({\mathcal R},{\mathcal S})\) (the space of all completely bounded maps from \({\mathcal R}\) to \({\mathcal S})\) is shown to be an isometry, and from this he deduces the result of Chatterjee and Smith that the natural map from the central Haagerup tensor product \({\mathcal R} \otimes_{\mathcal C} {\mathcal R}\) to \(\text{CB} ({\mathcal R}, {\mathcal R})\) is an isometry for each von Neumann algebra \({\mathcal R}\). He also shows that for an elementary operator on a prime \(C^*\)- algebra with zero socle or on a continuous von Neumann algebra the norm is equal to the completely bounded norm.
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 764029
zbMATH Keywords: Haagerup norm on the tensor product of submodules
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: injective
zbMATH Keywords: central Haagerup tensor product
zbMATH Keywords: elementary operator on a prime \(C^*\)-algebra
zbMATH Keywords: completely bounded norm
MaRDI profile type: Publication
full work available at URL: jfan.1995.1053
OpenAlex ID: W1979529801
DOI: 10.1006/JFAN.1995.1053
description: scientific article; zbMATH DE number 764029