Complemented subspaces of Banach spaces $C(K\times L)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916397531004928 |
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| author | Plebanek, Grzegorz Rondoš, Jakub Sobota, Damian |
| author_facet | Plebanek, Grzegorz Rondoš, Jakub Sobota, Damian |
| contents | We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(βω\timesβω)$ contains a complemented copy of $C([0,1]^κ)$ for every cardinal number $1\leκ\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(βω\timesβω)$ contains no complemented copy of $C_p(2^ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complemented subspaces of Banach spaces $C(K\times L)$ Plebanek, Grzegorz Rondoš, Jakub Sobota, Damian Functional Analysis Primary: 46B20, 46E15. Secondary: 28A33, 46B26 We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(βω\timesβω)$ contains a complemented copy of $C([0,1]^κ)$ for every cardinal number $1\leκ\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(βω\timesβω)$ contains no complemented copy of $C_p(2^ω)$. |
| title | Complemented subspaces of Banach spaces $C(K\times L)$ |
| topic | Functional Analysis Primary: 46B20, 46E15. Secondary: 28A33, 46B26 |
| url | https://arxiv.org/abs/2405.19120 |