It is proved 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 [oplus ] Y, then there exists a completely bounded automorphism τ: X [oplus ] Y → X [oplus ] Y with completely bounded inverse such that τZ = X0 [oplus ] Y0, where X0 and Y0 are completely complemented subspaces of X and Y, respectively. If X and Y are homogeneous, the existence is proved of such a τ under a weaker assumption that any operator from X to Y is strictly singular. An upper estimate is obtained for ∥τ∥cb∥τ−1∥cb if X and Y are separable homogeneous Hilbertian operator spaces. Also proved is the uniqueness of a ‘completely unconditional’ basis in X [oplus ] Y if X and Y satisfy certain conditions.