Banach spaces with arbitrary finite Baire order
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916945993924608 |
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| author | Pelczar-Barwacz, Anna Silber, Zdeněk Wawrzycki, Tomasz |
| author_facet | Pelczar-Barwacz, Anna Silber, Zdeněk Wawrzycki, Tomasz |
| contents | We investigate intrinsic Baire classes of Banach spaces defined by Argyros, Godefroy and Rosenthal (2003). We introduce a construction, for any Banach space $X$ with a basis, of an $\ell_1$-saturated separable Banach space $Y$ such that for any $α\leqslant ω_1$ we have $Y^{**}_{1+α} \cong Y \oplus X^{**}_α$, where $X^{**}_α$ denotes the $α$-th intrinsic Baire class of $X$. We apply this construction to answer two open problems by Argyros, Godefroy and Rosenthal (2003), namely we build separable Banach spaces of any Baire order less or equal to $ω$, and a non-universal separable Banach space of order $ω_1$. Finally, we apply the construction to show an analogue of a result of Lindenstrauss (1971) by constructing, for any Banach space $X$ with a basis and any $n \in \mathbb{N}$, a Banach space $Y$ such that $Y^{**}_n \cong Y^{**}_{n-1} \oplus X$, showing that any such $X$ can appear as the space of functionals in a bidual Banach space $Y^{**}$ that are of $n$-th intrinsic Baire class but not of $(n-1)$-th intrinsic Baire class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Banach spaces with arbitrary finite Baire order Pelczar-Barwacz, Anna Silber, Zdeněk Wawrzycki, Tomasz Functional Analysis 46B03, 46B10, 46B20, 54H05 We investigate intrinsic Baire classes of Banach spaces defined by Argyros, Godefroy and Rosenthal (2003). We introduce a construction, for any Banach space $X$ with a basis, of an $\ell_1$-saturated separable Banach space $Y$ such that for any $α\leqslant ω_1$ we have $Y^{**}_{1+α} \cong Y \oplus X^{**}_α$, where $X^{**}_α$ denotes the $α$-th intrinsic Baire class of $X$. We apply this construction to answer two open problems by Argyros, Godefroy and Rosenthal (2003), namely we build separable Banach spaces of any Baire order less or equal to $ω$, and a non-universal separable Banach space of order $ω_1$. Finally, we apply the construction to show an analogue of a result of Lindenstrauss (1971) by constructing, for any Banach space $X$ with a basis and any $n \in \mathbb{N}$, a Banach space $Y$ such that $Y^{**}_n \cong Y^{**}_{n-1} \oplus X$, showing that any such $X$ can appear as the space of functionals in a bidual Banach space $Y^{**}$ that are of $n$-th intrinsic Baire class but not of $(n-1)$-th intrinsic Baire class. |
| title | Banach spaces with arbitrary finite Baire order |
| topic | Functional Analysis 46B03, 46B10, 46B20, 54H05 |
| url | https://arxiv.org/abs/2509.08906 |