The Haagerup Norm on the Tensor Product of Operator Modules

It is proved that the (analogy of the) Haagerup norm on the tenser product of submodules of B(H) over a von Neumann algebra T ⊆ B(H) is injective. If R ⊆ S ⊆ B (H) are von Neumann algebras with S injective and T = R′ ∩ S, then the natural map fromS⊗TS S equipped with the Haagerup norm to CB(R, S) (the space of all completely bounded maps from R to S) is shown to be an isometry, and from this we deduce the result of Chattejee and Smith that the natural map from the central Haagerup tenser product R ⊗C R to CB(R, R) is an isometry for each von Neumann algebra R. 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.