Relative weak injectivity of operator system pairs

TL;DR

This paper introduces and studies the concept of relative weak injectivity in operator systems, extending known results from C*-algebras and characterizing it via the free group C*-algebra.

Contribution

It extends the characterization of relative weak injectivity from C*-algebras to operator systems using the free group C*-algebra.

Findings

Characterizes relative weak injectivity in operator systems.

Shows $C^*( ext{free group})$ characterizes this property.

Utilizes operator system tensor product theory.

Abstract

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg \cite{Kr} proved that the C*-algebra $C^*(\mathbb{f}_\infty)$ of the free group $\mathbb{f}_\infty$ on countably many generators characterizes relative weak injectivity for pairs of C*-algebras by means of the maximal tensor product. One of the main results of this paper shows that $C^*(\mathbb{f}_\infty)$ also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products \cite{KP1,KP2}.