On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map

TL;DR

This paper characterizes when the zero map is a boundary representation for non-unital operator systems and links approximation properties to the bidual being an injective von Neumann algebra, revealing structural insights.

Contribution

It provides a new characterization for boundary representations in non-unital operator systems and connects approximation properties to the injectivity of the bidual.

Findings

Zero map is a boundary representation under certain conditions.

Non-unital operator systems with approximation properties have injective biduals.

Such systems necessarily contain many positive elements.

Abstract

The purpose of this paper is two-fold: firstly, we give a characterization on the level of non-unital operator systems for when the zero map is a boundary representation. As a consequence, we show that a non-unital operator system arising from the direct limit of C*-algebras under positive maps is a C*-algebra if and only if its unitization is a C*-algebra. Secondly, we show that the completely positive approximation property and the completely contractive approximation property of a non-unital operator system is equivalent to its bidual being an injective von Neumann algebra. This implies in particular that all non-unital operator systems with the completely contractive approximation property must necessarily admit an abundance of positive elements.