The maximal operator space of a normed space

We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X), MAX(Y))=Gamma(2)*(X, Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of ''free'' unitaries.