De Finetti Theorems for the Unitary Dual Group
We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing R-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti th...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211721 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | De Finetti Theorems for the Unitary Dual Group. Isabelle Baraquin, Guillaume Cébron, Uwe Franz, Laura Maassen and Moritz Weber. SIGMA 18 (2022), 067, 29 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862740459209621504 |
|---|---|
| author | Baraquin, Isabelle Cébron, Guillaume Franz, Uwe Maassen, Laura Weber, Moritz |
| author_facet | Baraquin, Isabelle Cébron, Guillaume Franz, Uwe Maassen, Laura Weber, Moritz |
| citation_txt | De Finetti Theorems for the Unitary Dual Group. Isabelle Baraquin, Guillaume Cébron, Uwe Franz, Laura Maassen and Moritz Weber. SIGMA 18 (2022), 067, 29 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing R-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in *-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group ⁺ₙ. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in *-probability spaces. On the other hand, if we drop the assumption of faithful states in *-probability spaces, we obtain a non-trivial half of a de Finetti theorem similar to the case of the dual group action.
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| first_indexed | 2026-04-17T17:42:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211721 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:42:14Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Baraquin, Isabelle Cébron, Guillaume Franz, Uwe Maassen, Laura Weber, Moritz 2026-01-09T12:45:36Z 2022 De Finetti Theorems for the Unitary Dual Group. Isabelle Baraquin, Guillaume Cébron, Uwe Franz, Laura Maassen and Moritz Weber. SIGMA 18 (2022), 067, 29 pages 1815-0659 2020 Mathematics Subject Classification: 46L54; 46L65, 60G09 arXiv:2203.05852 https://nasplib.isofts.kiev.ua/handle/123456789/211721 https://doi.org/10.3842/SIGMA.2022.067 We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing R-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in *-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group ⁺ₙ. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in *-probability spaces. On the other hand, if we drop the assumption of faithful states in *-probability spaces, we obtain a non-trivial half of a de Finetti theorem similar to the case of the dual group action. M.W. is supported by SFB-TRR 195 and the DFG Heisenberg program. I.B. and U.F. are supported by an ANR project (No. ANR-19-CE40-0002). G.C. is supported by the Project MESA (ANR-18-CE40-006) and by the Project STARS (ANR-20-CE40-0008) of the French National Research Agency (ANR). We acknowledge the DAAD Procope program held by Roland Vergnioux and the fifth author from 2019–2020. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications De Finetti Theorems for the Unitary Dual Group Article published earlier |
| spellingShingle | De Finetti Theorems for the Unitary Dual Group Baraquin, Isabelle Cébron, Guillaume Franz, Uwe Maassen, Laura Weber, Moritz |
| title | De Finetti Theorems for the Unitary Dual Group |
| title_full | De Finetti Theorems for the Unitary Dual Group |
| title_fullStr | De Finetti Theorems for the Unitary Dual Group |
| title_full_unstemmed | De Finetti Theorems for the Unitary Dual Group |
| title_short | De Finetti Theorems for the Unitary Dual Group |
| title_sort | de finetti theorems for the unitary dual group |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211721 |
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