  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.
