Operator Space Structures on Banach Spaces

If X is a Banach space, then any linear isometry from X to B(H), for some Hilbert space H endows an operator space structure on X. Among these operator space structures admissible on X, two are of special importance, namely, the minimal operator space structure Min(X) and the maximal operator space structure Max(X). In this note, we present a characterization of these spaces up to complete isomorphism. We derive some conditions for a completely bounded linear bijection to become a complete isomorphism. Also, we introduce the notion of complete equivalence of operator space structures on a Banach space.