Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras
Let \(\mathcal{A}\) and \(\mathcal{B}\) be two factor von Neumann algebras. In this paper, we proved that a bijective mapping \(\Phi :\mathcal{A}\rightarrow \mathcal{B}\) satisfies \(\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}\) (where \(\circ \) is the special Jordan product...
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| Дата: | 2021 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2021
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1482 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-14822021-04-11T06:11:31Z Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras Ferreira, J. C. M. Marietto, M. G. B. \(\ast\)-ring isomorphisms, factor von Neumann algebras 47B48, 46L10 Let \(\mathcal{A}\) and \(\mathcal{B}\) be two factor von Neumann algebras. In this paper, we proved that a bijective mapping \(\Phi :\mathcal{A}\rightarrow \mathcal{B}\) satisfies \(\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}\) (where \(\circ \) is the special Jordan product on \(\mathcal{A}\) and \(\mathcal{B},\) respectively), for all elements \(a,b\in \mathcal{A}\), if and only if \(\Phi \) is a \(\ast \)-ring isomorphism. In particular, if the von Neumann algebras \(\mathcal{A}\) and \(\mathcal{B}\) are type I factors, then \(\Phi \) is a unitary isomorphism or a conjugate unitary isomorphism. Lugansk National Taras Shevchenko University 2021-04-10 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1482 10.12958/adm1482 Algebra and Discrete Mathematics; Vol 31, No 1 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1482/pdf Copyright (c) 2021 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
\(\ast\)-ring isomorphisms factor von Neumann algebras 47B48 46L10 |
| spellingShingle |
\(\ast\)-ring isomorphisms factor von Neumann algebras 47B48 46L10 Ferreira, J. C. M. Marietto, M. G. B. Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| topic_facet |
\(\ast\)-ring isomorphisms factor von Neumann algebras 47B48 46L10 |
| format |
Article |
| author |
Ferreira, J. C. M. Marietto, M. G. B. |
| author_facet |
Ferreira, J. C. M. Marietto, M. G. B. |
| author_sort |
Ferreira, J. C. M. |
| title |
Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| title_short |
Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| title_full |
Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| title_fullStr |
Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| title_full_unstemmed |
Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras |
| title_sort |
mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von neumann algebras |
| description |
Let \(\mathcal{A}\) and \(\mathcal{B}\) be two factor von Neumann algebras. In this paper, we proved that a bijective mapping \(\Phi :\mathcal{A}\rightarrow \mathcal{B}\) satisfies \(\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}\) (where \(\circ \) is the special Jordan product on \(\mathcal{A}\) and \(\mathcal{B},\) respectively), for all elements \(a,b\in \mathcal{A}\), if and only if \(\Phi \) is a \(\ast \)-ring isomorphism. In particular, if the von Neumann algebras \(\mathcal{A}\) and \(\mathcal{B}\) are type I factors, then \(\Phi \) is a unitary isomorphism or a conjugate unitary isomorphism. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2021 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1482 |
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AT ferreirajcm mappingspreservingsumofproductsacircbbaonfactorvonneumannalgebras AT mariettomgb mappingspreservingsumofproductsacircbbaonfactorvonneumannalgebras |
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2024-04-12T06:27:37Z |
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2024-04-12T06:27:37Z |
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