Irreducible operators in von Neumann algebras
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915650995224576 |
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| author | Adappa, Sukitha Ma, Minghui Shen, Junhao Shi, Rui Yang, Shanshan |
| author_facet | Adappa, Sukitha Ma, Minghui Shen, Junhao Shi, Rui Yang, Shanshan |
| contents | Let $\mathcal{M}$ be a separable von Neumann algebra with center $\mathcal{Z}(\mathcal{M})$. An operator $T$ in $\mathcal{M}$ is called irreducible if the von Neumann algebra $W^*(T)$ generated by $T$ has trivial relative commutant, i.e., $W^*(T)'\cap\mathcal{M}=\mathcal{Z}(\mathcal{M})$. In this paper, we show that irreducible operators in $\mathcal{M}$ form a norm-dense $G_δ$ set, which is a generalization of Halmos' theorem. Moreover, we prove that every operator in $\mathcal{M}$ is the sum of two irreducible operators, which is an analogue of Radjavi's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducible operators in von Neumann algebras Adappa, Sukitha Ma, Minghui Shen, Junhao Shi, Rui Yang, Shanshan Operator Algebras Let $\mathcal{M}$ be a separable von Neumann algebra with center $\mathcal{Z}(\mathcal{M})$. An operator $T$ in $\mathcal{M}$ is called irreducible if the von Neumann algebra $W^*(T)$ generated by $T$ has trivial relative commutant, i.e., $W^*(T)'\cap\mathcal{M}=\mathcal{Z}(\mathcal{M})$. In this paper, we show that irreducible operators in $\mathcal{M}$ form a norm-dense $G_δ$ set, which is a generalization of Halmos' theorem. Moreover, we prove that every operator in $\mathcal{M}$ is the sum of two irreducible operators, which is an analogue of Radjavi's theorem. |
| title | Irreducible operators in von Neumann algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2512.03448 |