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.

Grothendieck theorem, Operator spaces, Additive combinatorics
Discrete Analysis
Quantum and classical data transmission
This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/639.071.409 - Quantum and classical data transmission
Algorithms and Complexity

Briët, J, & Palazuelos, C. (2019). Failure of the trilinear operator space Grothendieck theorem. Discrete Analysis, (8). doi:10.19086/da.8805