Jordan norms for multilinear maps on C^*-algebras and Grothendieck’s inequalities - Christensen, Erik (MATHEMATICA SCANDINAVICA, 2024)
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 · Vert _J. The Jordan norm Φ Vert_J of a multilinear map is obtained via factorizations of Φ in the form Φ (a_1, … , a_n) = T_0 σ _1(a_1)T_1 … T_(n-1)σ _n(a_n)T_n , 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 Vert _J ≤ 2B Vert .