Operators with disconnected spectrum in von Neumann algebras
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911414508060672 |
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| author | Ma, Minghui Shi, Rui Wang, Tianze |
| author_facet | Ma, Minghui Shi, Rui Wang, Tianze |
| contents | Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $Φ$ a unitarily invariant $\|\cdot\|$-dominating norm on $\mathcal{I}$.
In this paper, we provide a necessary and sufficient condition on $Φ$ such that every operator in $\mathcal{M}$ can be expressed as the sum of an operator in $\mathcal{M}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose $Φ$-norm is arbitrarily small.
Similarly, if $\mathcal{A}$ is a unital $C^*$-algebra of real rank zero with dimension greater than one and $\mathcal{I}$ is an essential ideal in $\mathcal{A}$, then every element in $\mathcal{A}$ can be written as the sum of an operator in $\mathcal{A}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose norm is arbitrarily small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01424 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operators with disconnected spectrum in von Neumann algebras Ma, Minghui Shi, Rui Wang, Tianze Operator Algebras Primary 46L10, 47C15, Secondary 47B10, 47A58 Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $Φ$ a unitarily invariant $\|\cdot\|$-dominating norm on $\mathcal{I}$. In this paper, we provide a necessary and sufficient condition on $Φ$ such that every operator in $\mathcal{M}$ can be expressed as the sum of an operator in $\mathcal{M}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose $Φ$-norm is arbitrarily small. Similarly, if $\mathcal{A}$ is a unital $C^*$-algebra of real rank zero with dimension greater than one and $\mathcal{I}$ is an essential ideal in $\mathcal{A}$, then every element in $\mathcal{A}$ can be written as the sum of an operator in $\mathcal{A}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose norm is arbitrarily small. |
| title | Operators with disconnected spectrum in von Neumann algebras |
| topic | Operator Algebras Primary 46L10, 47C15, Secondary 47B10, 47A58 |
| url | https://arxiv.org/abs/2602.01424 |