The Nilpotency of the Nil Metric $\mathbb{F}$-Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911315868516352 |
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| author | de França, Antonio |
| author_facet | de França, Antonio |
| contents | Let $\mathbb{F}$ be a normed field. In this work, we prove that every nil complete metric $\mathbb{F}$-algebra is nilpotent when $\mathbb{F}$ has characteristic zero. This result generalizes Grabiner's Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric $\mathbb{F}$-algebra $\mathfrak{A}$ and its completion $C(\mathfrak{A})$ satisfy the same polynomial identities, and consequently, if $\mathsf{char}(\mathbb{F})=0$ and $C(\mathfrak{A})$ is nil, then $\mathfrak{A}$ is nilpotent. Our results allow us to resolve Köthe's Problem affirmatively for complete metric algebras over normed fields of characteristic zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Nilpotency of the Nil Metric $\mathbb{F}$-Algebras de França, Antonio Rings and Algebras Primary 16R10, Secondary 46H99, 16W99, 16W50, 16R40, 12J05 Let $\mathbb{F}$ be a normed field. In this work, we prove that every nil complete metric $\mathbb{F}$-algebra is nilpotent when $\mathbb{F}$ has characteristic zero. This result generalizes Grabiner's Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric $\mathbb{F}$-algebra $\mathfrak{A}$ and its completion $C(\mathfrak{A})$ satisfy the same polynomial identities, and consequently, if $\mathsf{char}(\mathbb{F})=0$ and $C(\mathfrak{A})$ is nil, then $\mathfrak{A}$ is nilpotent. Our results allow us to resolve Köthe's Problem affirmatively for complete metric algebras over normed fields of characteristic zero. |
| title | The Nilpotency of the Nil Metric $\mathbb{F}$-Algebras |
| topic | Rings and Algebras Primary 16R10, Secondary 46H99, 16W99, 16W50, 16R40, 12J05 |
| url | https://arxiv.org/abs/2504.17168 |