Isomorphisms of \(\mathcal{C}(K, E)\) spaces and height of \(K\) (Q6040037)

The present paper is dedicated to the isomorphic structure of Banach spaces of continuous functions, more precisely, to the vector-valued case of \(\mathcal{C}(K,E)\), where \(K\) is a compact topological space and \(E\) a Banach space. The results of the paper indicate that for sufficiently different compact spaces \(K_1\) and \(K_2\), the Banach spaces \(\mathcal{C}(K_1,E_1)\) and \(\mathcal{C}(K_2,E_2)\) are not isomorphic, regardless of \(E_1\) and \(E_2\). To put the results into context, recall that two Banach spaces \(\mathcal{C}(K_1)\) and \(\mathcal{C}(K_2)\) are isometric if and only if the compact spaces \(K_1\) and \(K_2\) are homeomorphic. Additionally, when \(K_1\) and \(K_2\) are not homeomorphic, the Banach-Mazur distance between \(\mathcal{C}(K_1)\) and \(\mathcal{C}(K_2)\) is at least 2 (the Amir-Cambern theorem). Moreover, \textit{Y. Gordon} [Isr. J. Math. 8, 391--397 (1970; Zbl 0205.12401)] proved that the Banach-Mazur distance between \(\mathcal{C}(K_1)\) and \(\mathcal{C}(K_2)\) is at least 3, whenever \(ht(K_1) \neq ht(K_2)\). In the light of these results, Pełczyński formulated the conjecture that the Banach-Mazur distance between two \(\mathcal{C}(K)\) spaces is always an integer number. In the vector-valued case, the situation is much less understood and many results have been obtained only very recently. For example, \textit{E. M. Galego} and \textit{M. Zahn} [J. Math. Anal. Appl. 431, No. 1, 622--632 (2015; Zbl 1343.46007)] proved that, for a uniformly convex Banach space \(E\) and metric spaces \(K_1\) and \(K_2\), \(\mathcal{C}(K_1,E)\) and \(\mathcal{C}(K_2,E)\) are isomorphic if and only if \(\mathcal{C}(K_1)\) is isomorphic to \(\mathcal{C}(K_2)\). One of the main outcomes of the paper consists in substantially relaxing the uniform convexity assumption on \(E\). In order to state the main result of the paper, we need one piece of notation. For an ordinal \(\alpha\), denote by \(\Gamma(\alpha)\) the least ordinal not less than \(\alpha\) that has the form \(\omega^\eta\) (for some ordinal \(\eta\)). Such an ordinal function appears naturally in this context, as \(\Gamma(ht(K_1))= \Gamma(ht(K_2))\) is a necessary condition for \(\mathcal{C}(K_1)\) to be isomorphic to \(\mathcal{C}(K_2)\) (which is also sufficient when \(K_1\) and \(K_2\) are metrisable). The main result of the paper is then the following: Assume that \(E_1\) and \(E_2\) are Banach spaces that do not contain a copy of \(c_0\) and \(K_1\) and \(K_2\) satisfy \(ht(K_2)\leq\omega^\alpha (k+1)\) and \(ht(K_1)> \omega^\alpha n\), for some \(n>k\). Then \[ d_{BM}\left(\mathcal{C}(K_1,E_1), \mathcal{C}(K_2,E_2)\right) \geq \max\left\{3, \frac{2n-k}{k}\right\}. \] In particular, under the same assumptions, one gets the following: \begin{itemize} \item[1.] If \(\mathcal{C}(K_1,E_1)\) is isomorphic to \(\mathcal{C}(K_2,E_2)\), then \(\Gamma(ht(K_1))= \Gamma(ht(K_2))\). \item[2.] If additionally \(K_1\) and \(K_2\) are metric spaces, then \(\mathcal{C}(K_1,E)\) is isomorphic to \(\mathcal{C}(K_2,E)\) if and only if \(\mathcal{C}(K_1)\) is isomorphic to \(\mathcal{C}(K_2)\). \end{itemize}