On a question of Blecher, Pisier, Shlyakhtenko

We show the failure of a matricial version of Grothendieck's theorem for operator spaces, thereby resolving a long-standing open question in the field. Moreover, by showing that such a counterexample can occur in the simplest context of commutative $C^*$-algebras, we address some other open questions in operator algebras. Our constructions, completely explicit and fairly simple, are inspired by some techniques in quantum information theory.