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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.11075 |
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| _version_ | 1866914801741987840 |
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| author | Mastnak, Mitja Radjavi, Heydar |
| author_facet | Mastnak, Mitja Radjavi, Heydar |
| contents | We consider the following question: Let $\mathcal{A}$ be an abelian self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{H}$. Assume that $\mathcal{A}$ is invariant under conjugation by a unitary operator $U$, i.e., $U^* AU$ is in $\mathcal{A}$ for every member $A$ of $\mathcal{A}$. Is there a maximal abelian self-adjoint algebra containing $\mathcal{A}$, which is still invariant under conjugation by $U$? The answer, which is easily seen to be yes in finite dimensions, is not trivial in general. We prove affirmative answers in special cases including the one where $\mathcal{A}$ is generated by a compact operator. We also construct a counterexample in the general case, whose existence is perhaps surprising. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant embeddings and ergodic obstructions Mastnak, Mitja Radjavi, Heydar Functional Analysis 47L30, 46L40, 47D03 We consider the following question: Let $\mathcal{A}$ be an abelian self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{H}$. Assume that $\mathcal{A}$ is invariant under conjugation by a unitary operator $U$, i.e., $U^* AU$ is in $\mathcal{A}$ for every member $A$ of $\mathcal{A}$. Is there a maximal abelian self-adjoint algebra containing $\mathcal{A}$, which is still invariant under conjugation by $U$? The answer, which is easily seen to be yes in finite dimensions, is not trivial in general. We prove affirmative answers in special cases including the one where $\mathcal{A}$ is generated by a compact operator. We also construct a counterexample in the general case, whose existence is perhaps surprising. |
| title | Invariant embeddings and ergodic obstructions |
| topic | Functional Analysis 47L30, 46L40, 47D03 |
| url | https://arxiv.org/abs/2405.11075 |