OPERATOR SPACES AMS Subject Classification : 46 L 07 , 47 L 25 Timur

We prove that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X ⊕ Y , then there exists a completely bounded automorphism τ : X ⊕ Y → X ⊕ Y with completely bounded inverse such that τZ = X0 ⊕ Y0, where X0 and Y0 are completely complemented subspaces of X and Y , respectively. If X and Y are homogeneous, we prove the existence of such a τ under a weaker assumption that any operator operator from X to Y is strictly singular. We are able to obtain an upper estimate for ||τ ||cb||τ||cb if X and Y are separable homogeneous Hilbertian operator spaces. We also prove the uniqueness of a “completely unconditional” basis in X ⊕ Y if X and Y satisfy certain conditions.