Operator ideals and operator spaces

We prove that every full symmetrically normed ideal of operators on a Hilbert space is realizable as the set of completely bounded maps between two homogeneous operator Hilbert spaces, with the c.b. norm equivalent to (but in general not equal to) the symmetric norm. We show that one can have equality of the c.b. norm and the symmetric norm if one leaves the category of operator spaces and passes to a slightly larger category.