# Failure of the trilinear operator space Grothendieck theorem

## TL;DR

This paper provides a counterexample demonstrating that a trilinear operator space Grothendieck theorem does not hold universally, revealing limitations in the relationship between certain norms in operator space theory.

## Contribution

The authors construct a counterexample showing the unbounded ratio between symmetrized completely bounded and jointly completely bounded norms for trilinear forms on , answering a key open question.

## Key findings

- The ratio of norms is unbounded for certain trilinear forms.
- Counterexample disproves the universal validity of the trilinear operator space Grothendieck theorem.
- The proof employs a non-commutative generalized von Neumann inequality.

## Abstract

We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on $\ell_\infty$, 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.