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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.06127
- arXiv:
- arXiv:2408.06127
- Bibcode:
- 2024arXiv240806127K
- Keywords:
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- Mathematics - Operator Algebras