A vector-valued Grothendieck inequality with an application to (p,q)-completely bounded operators

A vector-valued version of Grothendieck's inequality (which seems to be of independent interest) is used to show that for operators between operator s paces the completely bounded norm and the so-called (infinity,1)-completely bounded norm are equivalent - the latter norm is a priori larger than the cb-norm and the largest among the scale of all (q,p)-completely bounded nor ms (for q = p originally invented by Pisier), The link between these two Gr othendieck type inqualities is a vector-valued version of the Maurey-Rosent hal factorization theorem.