Products of irreducible operators in factors
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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_ | 1866918248005500928 |
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| author | Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze |
| author_facet | Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze |
| contents | Let $\mathcal M$ be a separable factor.
An operator $T$ in $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann algebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$.
In this paper, we show that every operator in a separable factor $\mathcal{M}$ is the product of two irreducible operators in $\mathcal{M}$, except the zero operator in factors of type $\mathrm{I}_{2n+1}$ for $n\geqslant 1$.
This may be viewed as a multiplicative analogue of Radjavi's result which asserts that every operator on a separable Hilbert space is the sum of two irreducible operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12162 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Products of irreducible operators in factors Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze Operator Algebras Functional Analysis Primary 46L10, Secondary 47C15 Let $\mathcal M$ be a separable factor. An operator $T$ in $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann algebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$. In this paper, we show that every operator in a separable factor $\mathcal{M}$ is the product of two irreducible operators in $\mathcal{M}$, except the zero operator in factors of type $\mathrm{I}_{2n+1}$ for $n\geqslant 1$. This may be viewed as a multiplicative analogue of Radjavi's result which asserts that every operator on a separable Hilbert space is the sum of two irreducible operators. |
| title | Products of irreducible operators in factors |
| topic | Operator Algebras Functional Analysis Primary 46L10, Secondary 47C15 |
| url | https://arxiv.org/abs/2512.12162 |