Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866914954416750592 |
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| author | Dimant, Verónica Lassalle, Silvia Maestre, Manuel |
| author_facet | Dimant, Verónica Lassalle, Silvia Maestre, Manuel |
| contents | In the early nineties, R. M. Aron, B. Cole, T. Gamelin and W.B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $\ell_p$ ($1\le p<\infty$). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $p\in\mathbb N$ ($p\ge 2$), the fiber of every $z\in B_{\ell_p}$ contains a set of cardinal $2^{\mathfrak c}$ such that any two elements of this set belong to different Gleason parts. For the case $p=1$, we complete the known description of the fibers, showing that, for each $z\in\overline B_{\ell_1''}\setminus S_{\ell_1}$, the fiber over $z$ is not a singleton. Also, we establish that different fibers over elements in $S_{\ell_1''}$ cannot share Gleason parts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$ Dimant, Verónica Lassalle, Silvia Maestre, Manuel Functional Analysis Complex Variables 46J15, 46E50, 46G20 In the early nineties, R. M. Aron, B. Cole, T. Gamelin and W.B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $\ell_p$ ($1\le p<\infty$). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $p\in\mathbb N$ ($p\ge 2$), the fiber of every $z\in B_{\ell_p}$ contains a set of cardinal $2^{\mathfrak c}$ such that any two elements of this set belong to different Gleason parts. For the case $p=1$, we complete the known description of the fibers, showing that, for each $z\in\overline B_{\ell_1''}\setminus S_{\ell_1}$, the fiber over $z$ is not a singleton. Also, we establish that different fibers over elements in $S_{\ell_1''}$ cannot share Gleason parts. |
| title | Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$ |
| topic | Functional Analysis Complex Variables 46J15, 46E50, 46G20 |
| url | https://arxiv.org/abs/2409.13889 |