The Haagerup norm on the tensor product of operator modules
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.
Haagerup norm on the tensor product of submodules
elementary operator on a prime \(C^*\)-algebra
scientific article; zbMATH DE number 764029