Operator spaces with few completely bounded maps

We construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable 1-Hilbertian operator space X₀ such that, whenever X′ is an infinite dimensional quotient of X₀, X is a subspace of X′, and T : X→X′ is a completely bounded map, then T = λI _X + S, where S is compact Hilbert-Schmidt and ||S||₂/16 ≤ ||S||_cb ≤ ||S||₂. Moreover, every infinite dimensional quotient of a subspace of X₀fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property.