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, answering a question
of Pisier. The proof is based on a non-commutative version of the generalized
von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi