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
dx.doi.org/10.19086/da.8805
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