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We give a counterexample to a trilinear version of the operator spaceGrothendieck theorem. In particular, we show that for trilinear forms on$\ell_\infty$, the ratio of the symmetrized completely bounded norm and thejointly completely bounded norm is in general unbounded, answering a questionof Pisier. The proof is based on a non-commutative version of the generalizedvon Neumann inequality from additive combinatorics.

Failure of the trilinear operator space Grothendieck theorem