Weak* decomposition and Radon-Nikodym theorem for quantum expectations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908447052660736 |
|---|---|
| author | Naderi, Fouad |
| author_facet | Naderi, Fouad |
| contents | A quantum expectation is a positive linear functional of norm one on a non-commutative probability space (i.e., a C*-algebra). For a given pair of quantum expectations $μ$ and $λ$ on a non-commutative probability space $A$, we propose a definition for weak* continuity and weak* singularity of $μ$ with respect to $λ$. Then, using the theory of von Neumann algebras, we obtain the natural weak* continuous and weak* singular parts of $μ$ with respect to $λ$. If $λ$ satisfies a weak tracial property known as the KMS condition, we show that our weak* decomposition coincides with the Arveson-Gheondea-Kavruk Lebesgue (AGKL) decomposition. This equivalence allows us to compute the Radon-Nikodym derivative of $μ$ with respect to $λ$. We also discuss the possibility of extending our results to the positive linear functionals defined on the Cuntz-Toeplitz operator system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak* decomposition and Radon-Nikodym theorem for quantum expectations Naderi, Fouad Operator Algebras A quantum expectation is a positive linear functional of norm one on a non-commutative probability space (i.e., a C*-algebra). For a given pair of quantum expectations $μ$ and $λ$ on a non-commutative probability space $A$, we propose a definition for weak* continuity and weak* singularity of $μ$ with respect to $λ$. Then, using the theory of von Neumann algebras, we obtain the natural weak* continuous and weak* singular parts of $μ$ with respect to $λ$. If $λ$ satisfies a weak tracial property known as the KMS condition, we show that our weak* decomposition coincides with the Arveson-Gheondea-Kavruk Lebesgue (AGKL) decomposition. This equivalence allows us to compute the Radon-Nikodym derivative of $μ$ with respect to $λ$. We also discuss the possibility of extending our results to the positive linear functionals defined on the Cuntz-Toeplitz operator system. |
| title | Weak* decomposition and Radon-Nikodym theorem for quantum expectations |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2506.12018 |