Jordan norms for multilinear maps on C^-algebras and Grothendieck's inequalities

There exists a generalization of the concept completely bounded norm, for multilinear maps on C-algebras. We will use the word Jordan norm, for this norm and denote it by · J. The Jordan norm ΦJ of a multilinear map is obtained via factorizations of Φ in the form Φ (a₁, , aₙ) = T₀ σ ₁(a₁)T₁ ⋯ T₍ₙ₋₁₎σ ₙ(aₙ)Tₙ , where the maps σ ᵢ are Jordan homomorphisms. We show that any bounded bilinear form on a pair of C-algebras is Jordan bounded and satisfies B J ≤ 2 B.