Operator system structures on the unital direct sum of C*-algebras

TL;DR

This paper investigates operator system structures on the unital sum of C*-algebras, revealing their interrelations and improving bounds related to tensor norms, with implications for quantum entanglement and the QWEP conjecture.

Contribution

It introduces and compares three natural operator system structures on the unital sum of C*-algebras, including the coproduct, and applies these to improve bounds on tensor norms.

Findings

Identifies the coproduct as a key operator system structure.

Establishes interrelations between different operator system structures.

Provides improved bounds on C*-tensor norms, impacting quantum information theory.

Abstract

This work is motivated by Radulescu's result on the comparison of C*-tensor norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural inclusions of A and B into their unital free product, their maximal tensor product and their minimal tensor product. These inclusions define three operator system structures on the internal sum A+B, the first of which we identify as the coproduct of A and B in the category of operator systems. Partly using ideas from quantum entanglement theory, we prove various interrelations between these three operator systems. As an application, the present results yield a significant improvement over Radulescu's bound on C*(F_n) x C*(F_n). At the same time, this tight comparison is so general that it cannot be regarded as evidence for a positive answer to the QWEP conjecture.