Isomorphisms of $\mathcal{C}(K, E)$ spaces and height of $K$
Abstract
Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ based on the ordinals $ht(K_1)$, $ht(K_2)$, which are new even for the case of spaces of real valued functions on ordinal intervals. As a corollary we deduce that $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ are not isomorphic if $ht(K_1)$ is substantially different from $ht(K_2)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2022
- DOI:
- 10.48550/arXiv.2206.09137
- arXiv:
- arXiv:2206.09137
- Bibcode:
- 2022arXiv220609137R
- Keywords:
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- Mathematics - Functional Analysis;
- 46E15;
- 46B03;
- 47B38;
- 54D30