Jordan norms for multilinear maps on $C^{\ast}$-algebras and Grothendieck's inequalities


  • Erik Christensen



There exists a generalization of the concept completely bounded norm, for multilinear maps on $C^{\ast }$-algebras. We will use the word Jordan norm, for this norm and denote it by $\lVert \cdot \rVert _J$. 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 $.


Alfsen, E. M., and Shultz, F. W. Geometry of state spaces of operator algebras, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 2003.

Christensen, E., Bilinear forms, Schur multipliers, complete boundedness and duality, Math. Scand. 129 (2023), no. 3, 543–569.

Christensen, E., Some points of view on Grothendieck's inequalities, Submitted, arXiv:2312.09029.

Christensen, E., and Sinclair, A. M., Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), no. 1, 151–181.

Haagerup, U., The Grothendieck inequality for bilinear forms on $C^* $-algebras, Adv. in Math. 56 (1985), no. 2, 93–116.

Haagerup, U., and Musat, M., The Effros-Ruan conjecture for bilinear forms on $C^*$-algebras, Invent. Math. 174 (2008), no. 1, 139–163.

Kadison, R. V., Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338.

Kadison, R. V., and Ringrose, J. R., Fundamentals of the theory of operator algebras, Pure and Applied Mathematics, 100. Academic Press, Inc., Orlando, FL, 1986.

Paulsen, V. I., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.

Pisier, G., Grothendieck's theorem for non commutative $C^*$-algebras, with an appendix on Grothendieck's constants, J. Funct. Anal. 29 (1978), no. 3, 397–415.

Pisier, G., Grothendieck's theorem, past and present, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 2, 237–323.

Pisier, G., and Shlyakthenko, D., Grothendieck's theorem for operator spaces, Invent. Math. 150 (2002), no. 1, 185–217.

Stinespring, W. F., Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.

Takesaki, M., Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128. Springer-Verlag, Berlin-New York, 1970.



How to Cite

Christensen, E. (2024). Jordan norms for multilinear maps on $C^{\ast}$-algebras and Grothendieck’s inequalities. MATHEMATICA SCANDINAVICA, 130(2).