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 subset of or equal to B(H) is injective. If R subset of or equal to L subset of or equal to B (H) are von Neumann algebras with L injective and T = R' boolean AND L, then the natural map from L x(T), L equipped with the Haagerup norm to CB(R, L) (the space of all completely bounded maps from R to L) 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 x(l), 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. (C) 1995 Academic Press, Inc.