Absolutely dilatable bimodule maps
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913806274265088 |
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| author | Chatzinikolaou, Alexandros Todorov, Ivan G. Turowska, Lyudmila |
| author_facet | Chatzinikolaou, Alexandros Todorov, Ivan G. Turowska, Lyudmila |
| contents | We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Absolutely dilatable bimodule maps Chatzinikolaou, Alexandros Todorov, Ivan G. Turowska, Lyudmila Operator Algebras Functional Analysis 47A20 (Primary), 46L07, 46M05 (Secondary) We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum. |
| title | Absolutely dilatable bimodule maps |
| topic | Operator Algebras Functional Analysis 47A20 (Primary), 46L07, 46M05 (Secondary) |
| url | https://arxiv.org/abs/2411.08086 |