On a variant of the Grothendieck inequality and estimates on tensor product norms

In this paper we propose a generalization of the Grothendieck inequality for pairs of Banach spaces E and F with E being finite dimensional and investigate the behaviour of the Grothendieck constant KG(E,F) implicit in such an inequality. We show that if sup{KG(En,F):n?1} is finite for some sequence of finite dimensional Banach spaces (En)n?1 with dimEn=n, and an infinite dimensional Banach space F, then both F and F? must have finite cotype. In addition to that if F has the bounded approximation property, we conclude that (E?n)n?1 satisfies G.T. uniformly by assuming the validity of a conjecture due to Pisier. We also show that KG(E,F) is closely related to the constant ?(E,F), introduced recently, comparing the projective and injective norms on the tensor product of two finite dimensional Banach spaces E and F. We also study, analogously, these constants by computing the supremum only on non-negative tensors.