Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910051818536960 |
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| author | Rondoš, Jakub Sobota, Damian |
| author_facet | Rondoš, Jakub Sobota, Damian |
| contents | For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the product $E \times E^*$, based on new descriptions of the Banach spaces $\mathcal{C}([1, ω^α])$ for countable ordinal numbers $α$ and $\mathcal{C}(2^ω)$. Applying this general result in the case where $E=\mathcal{C}(K)$ for some compact space $K$, we further obtain a characterization of the existence of a positively $1$-complemented positively isometric copy of $\mathcal{C}(L)$ inside $\mathcal{C}(K)$ in terms of the topology of $K$ and the space of probability Radon measures on $K$. In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces Rondoš, Jakub Sobota, Damian Functional Analysis 46E15, 46B03, 46B04 (Primary) 47B38, 54A25, 54C35 (Secondary) For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the product $E \times E^*$, based on new descriptions of the Banach spaces $\mathcal{C}([1, ω^α])$ for countable ordinal numbers $α$ and $\mathcal{C}(2^ω)$. Applying this general result in the case where $E=\mathcal{C}(K)$ for some compact space $K$, we further obtain a characterization of the existence of a positively $1$-complemented positively isometric copy of $\mathcal{C}(L)$ inside $\mathcal{C}(K)$ in terms of the topology of $K$ and the space of probability Radon measures on $K$. In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces. |
| title | Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces |
| topic | Functional Analysis 46E15, 46B03, 46B04 (Primary) 47B38, 54A25, 54C35 (Secondary) |
| url | https://arxiv.org/abs/2603.12922 |