On a question of Blecher, Pisier, Shlyakhtenko

TL;DR

This paper demonstrates the failure of a matricial version of Grothendieck's theorem in operator spaces, providing explicit counterexamples in commutative $C^*$-algebras and connecting techniques from quantum information theory.

Contribution

It resolves a long-standing open question by showing the theorem's failure in a simple, commutative setting with explicit constructions inspired by quantum information theory.

Findings

Counterexample to matricial Grothendieck's theorem in operator spaces

Failure occurs even in commutative $C^*$-algebras

Explicit, simple constructions based on quantum information techniques

Abstract

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.