Jordan norms for multilinear maps on C∗-algebras and Grothendieck's inequalities
DOI:
https://doi.org/10.7146/math.scand.a-143520Abstract
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 ‖. The Jordan norm \lVert\Phi\rVert_J of a multilinear map is obtained via factorizations of \Phi in the form \Phi (a_1, \dots , a_n) = T_0 \sigma _1(a_1)T_1 \cdots T_{(n-1)}\sigma _n(a_n)T_n , where the maps \sigma _i are Jordan homomorphisms. We show that any bounded bilinear form on a pair of C^{\ast }-algebras is Jordan bounded and satisfies \lVert B\rVert _J \leq 2\lVert B\rVert .
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