Ortho-isomorphisms of von Neumann algebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918229484503040 |
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| author | Ma, Minghui Shi, Weijuan |
| author_facet | Ma, Minghui Shi, Weijuan |
| contents | Suppose $\mathscr M$ and $\mathscr N$ are von Neumann algebras. Two operators $A$ and $B$ in $\mathscr M$ are said to be orthogonal if $A^*B=0$, meaning their ranges are orthogonal. Let $φ\colon\mathscr M\to\mathscr N$ be a map. We say that $φ$ is an ortho-isomorphism if it is bijective and satisfies that $A^*B=0$ if and only if $φ(A)^*φ(B)=0$ for all $A,B\in\mathscr M$. The map $φ$ is called ortho-additive if the additive relation $φ(A+B)=φ(A)+φ(B)$ holds for all $A,B\in \mathscr M$ with $A^*B=0$. In this paper, we characterize the complete structure of ortho-additive ortho-isomorphisms between von Neumann algebras, which is an analogue of Dye's theorem and Uhlhorn's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14697 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ortho-isomorphisms of von Neumann algebras Ma, Minghui Shi, Weijuan Operator Algebras Suppose $\mathscr M$ and $\mathscr N$ are von Neumann algebras. Two operators $A$ and $B$ in $\mathscr M$ are said to be orthogonal if $A^*B=0$, meaning their ranges are orthogonal. Let $φ\colon\mathscr M\to\mathscr N$ be a map. We say that $φ$ is an ortho-isomorphism if it is bijective and satisfies that $A^*B=0$ if and only if $φ(A)^*φ(B)=0$ for all $A,B\in\mathscr M$. The map $φ$ is called ortho-additive if the additive relation $φ(A+B)=φ(A)+φ(B)$ holds for all $A,B\in \mathscr M$ with $A^*B=0$. In this paper, we characterize the complete structure of ortho-additive ortho-isomorphisms between von Neumann algebras, which is an analogue of Dye's theorem and Uhlhorn's theorem. |
| title | Ortho-isomorphisms of von Neumann algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2509.14697 |