Some points of view on Grothendieck's inequalities

Haagerup’s proof of the non commutative little Grothendieck
inequality raises some questions on the commutative little
inequality, and it offers a new result on scalar matrices with
non negative entries. The theory of completely bounded maps
may be used to show that the commutative Grothendieck
inequality follows from the little commutative inequality, and
that this passage may be given a geometric form as a relation
between a pair of compact convex sets of positive matrices,
which, in turn, characterizes the little constant k^C_G.