  We construct a family of separable Hilbertian operator spaces, such that the
relation of complete isomorphism between the subspaces of each member of this
family is complete $\ks$. We also investigate some interesting properties of
completely unconditional bases of the spaces from this family. In the Banach
space setting, we construct a space for which the relation of isometry of
subspaces is equivalent to equality of real numbers.
