On the central Haagerup tensor product and completely bounded mappings of a C*-algebra

We consider the natural contractive map from the central Haagerup tensor product of a unital C*-algebra A with itself to the space of completely bounded maps CB(A) on A. We establish the necessity of the known sufficient condition for isometry of the map, namely that all Glimm ideals of A are primal. However, when the map is restricted to tensors with length bounded by a fixed quantity, a weaker necessary and sufficient condition is established.