The Haagerup Norm on the Tensor Product of Operator Modules

✍ Scribed by B. Magajna

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 989 KB
• Volume: 129
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1995.1053

✦ Synopsis

It is proved that the (analogy of the) Haagerup norm on the tensor product of submodules of (\mathscr{B}(\mathscr{H})) over a von Neumann algebra (\mathscr{T} \subseteq \mathscr{B}(\mathscr{C})) is injective. If (\mathscr{A} \subseteq \mathscr{S} \subseteq \mathscr{A}(\mathscr{H})) are von Neumann algebras with (\mathscr{S}) injective and (\mathscr{Z}=\mathscr{A}^{\prime} \cap \mathscr{F}), then the natural map from (\mathscr{S} \otimes, \mathscr{S}) equipped with the Haagerup norm to CB( (\mathscr{X}, \mathscr{S})) (the space of all completely bounded maps from (\mathscr{R}) to (\mathscr{S}^{\prime}) ) is shown to be an isometry, and from this we deduce the result of Chatterjee and Smith that the natural map from the central Haagerup tensor product (A \otimes \otimes_{6}) to (C B(R, M)) is an isometry for each von Neumann algebra 3 . It is also shown 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. 1995 Academic Press. Inc.