Searching for linear structures in the failure of the Stone-Weierstrass theorem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909283836231680 |
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| author | Caballer, Marc Dantas, Sheldon Rodríguez-Vidanes, Daniel L. |
| author_facet | Caballer, Marc Dantas, Sheldon Rodríguez-Vidanes, Daniel L. |
| contents | We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal{C}(L, \mathbb{K})$ that vanishes nowhere on $L$ but does not necessarily separate the points of $L$. We address the problem of finding the precise codimension of $\overline{\mathcal{A}}$ in a broad setting, which allows us to describe the lineability of $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $(\mathcal{C}(\partial{D}, \mathbb{C}) \setminus \overline{\text{Pol}(\partial{D})}) \cup \{0\}$, where $\text{Pol}(\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Searching for linear structures in the failure of the Stone-Weierstrass theorem Caballer, Marc Dantas, Sheldon Rodríguez-Vidanes, Daniel L. Functional Analysis We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal{C}(L, \mathbb{K})$ that vanishes nowhere on $L$ but does not necessarily separate the points of $L$. We address the problem of finding the precise codimension of $\overline{\mathcal{A}}$ in a broad setting, which allows us to describe the lineability of $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $(\mathcal{C}(\partial{D}, \mathbb{C}) \setminus \overline{\text{Pol}(\partial{D})}) \cup \{0\}$, where $\text{Pol}(\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account. |
| title | Searching for linear structures in the failure of the Stone-Weierstrass theorem |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2405.06453 |