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arXiv · 2408.06127

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

Se-Jin Kim
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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.

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Se-Jin Kim. 2024-08-12. On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map. https://arxiv.org/abs/2408.06127

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