On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map
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.
Keywords
Cite
@article{arxiv.2408.06127,
title = {On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map},
author = {Se-Jin Kim},
journal= {arXiv preprint arXiv:2408.06127},
year = {2024}
}