We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on ℓ∞, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.

doi.org/10.19086/da.8805
arXiv.org e-Print archive
Quantum and classical data transmission
Algorithms and Complexity

Briët, J, & Palazuelos, C. (2018). Failure of the trilinear operator space Grothendieck theorem. arXiv.org e-Print archive. doi:10.19086/da.8805