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.

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Discrete Analysis
Quantum and classical data transmission
Algorithms and Complexity

Briët, J., & Palazuelos, C. (2019). Failure of the trilinear operator space Grothendieck theorem. Discrete Analysis, 2019, 8:1–8:16. doi:10.19086/da.8805