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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.14102 |
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| _version_ | 1866909038888878080 |
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| author | Avilés, Antonio Langemets, Johann Martín, Miguel Zoca, Abraham Rueda |
| author_facet | Avilés, Antonio Langemets, Johann Martín, Miguel Zoca, Abraham Rueda |
| contents | We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of examples and results. First, we characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. Besides, the perfect Daugavet property characterizes the absence of $G_δ$-points in $K$. We also study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces, with a number of applications. We classify these properties for $L_1(μ)$ and $L_\infty(μ)$ spaces in terms of the Maharam's decomposition of the measure. We also show that the space of Lipschitz functions $\Lip(M)$ on a complete length metric space has the $ω$-perfect Daugavet property, improving the previous knowledge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transfinite Daugavet property Avilés, Antonio Langemets, Johann Martín, Miguel Zoca, Abraham Rueda Functional Analysis We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of examples and results. First, we characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. Besides, the perfect Daugavet property characterizes the absence of $G_δ$-points in $K$. We also study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces, with a number of applications. We classify these properties for $L_1(μ)$ and $L_\infty(μ)$ spaces in terms of the Maharam's decomposition of the measure. We also show that the space of Lipschitz functions $\Lip(M)$ on a complete length metric space has the $ω$-perfect Daugavet property, improving the previous knowledge. |
| title | Transfinite Daugavet property |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2604.14102 |