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 $\lVert · \rVert _J$. The Jordan norm $\lVertΦ\rVert_J$ 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 $\lVert B\rVert _J ≤ 2\lVert B\rVert $.