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 is a completely bounded map, then T=I_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 X₀ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property. Mathematics Subject Classification (2000):The first author was supported in part by the NSF grants DMS-9970369, 0296094, and 0200714.