Essentially Commuting with a Unitary
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915093639331840 |
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| author | Chung, Jui-Hui Shapiro, Jacob |
| author_facet | Chung, Jui-Hui Shapiro, Jacob |
| contents | Let $R$ be a unitary operator whose spectrum is the circle. We show that the set of unitaries $U$ which essentially commute with $R$ (i.e., $[U,R]\equiv UR-RU$ is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with $R$ and obey a non-triviality condition, and prove it is bijective with $\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03934 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Essentially Commuting with a Unitary Chung, Jui-Hui Shapiro, Jacob Functional Analysis Mathematical Physics Operator Algebras Let $R$ be a unitary operator whose spectrum is the circle. We show that the set of unitaries $U$ which essentially commute with $R$ (i.e., $[U,R]\equiv UR-RU$ is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with $R$ and obey a non-triviality condition, and prove it is bijective with $\mathbb{Z}$. |
| title | Essentially Commuting with a Unitary |
| topic | Functional Analysis Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2501.03934 |