Unique matrix factorizations associated to bilinear forms and Schur multipliers

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m×n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.