Reducible operators in non-$Γ$ type ${\rm II}_1$ factors
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2019
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918177914486784 |
|---|---|
| author | Shen, Junhao Shi, Rui |
| author_facet | Shen, Junhao Shi, Rui |
| contents | A famous question of Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. In [30], Voiculescu gave this problem an affirmative answer by his remarkable non-commutative Weyl-von Neumann theorem. We investigate the existence or non-existence of an analogue of Voiculescu's result in factors of type ${\rm II}_1$.
In the paper, we prove that, in the operator norm topology, the set of reducible operators is ${\it nowhere}$ dense in a non-$Γ$ factor $\mathcal M$ of type ${\rm II}_1$, where separable and non-separable cases of $\mathcal M$ are both considered. Main tools developed in the paper are a new characterization of Murray and von Neumann's Property $Γ$ for a factor of type ${\rm II}_1$ and a spectral gap property for a single operator in a non-$Γ$ factor of type ${\rm II}_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_00573 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Reducible operators in non-$Γ$ type ${\rm II}_1$ factors Shen, Junhao Shi, Rui Operator Algebras 46L10, 47C15 A famous question of Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. In [30], Voiculescu gave this problem an affirmative answer by his remarkable non-commutative Weyl-von Neumann theorem. We investigate the existence or non-existence of an analogue of Voiculescu's result in factors of type ${\rm II}_1$. In the paper, we prove that, in the operator norm topology, the set of reducible operators is ${\it nowhere}$ dense in a non-$Γ$ factor $\mathcal M$ of type ${\rm II}_1$, where separable and non-separable cases of $\mathcal M$ are both considered. Main tools developed in the paper are a new characterization of Murray and von Neumann's Property $Γ$ for a factor of type ${\rm II}_1$ and a spectral gap property for a single operator in a non-$Γ$ factor of type ${\rm II}_1$. |
| title | Reducible operators in non-$Γ$ type ${\rm II}_1$ factors |
| topic | Operator Algebras 46L10, 47C15 |
| url | https://arxiv.org/abs/1907.00573 |