Embeddings of $C(K)$ spaces into $C(S, X)$ spaces with distortion strictly less than 3
Volume 220 / 2013
Abstract
In the spirit of the classical Banach–Stone theorem, we prove that if $K$ and $S$ are intervals of ordinals and $X$ is a Banach space having non-trivial cotype, then the existence of an isomorphism $T$ from $C(K, X)$ onto $C(S, X)$ with distortion $\|T\| \, \|T^{-1}\|$ strictly less than 3 implies that some finite topological sum of $K$ is homeomorphic to some finite topological sum of $S$. Moreover, if $X^{n}$ contains no subspace isomorphic to $X^{n+1}$ for every $n \in \mathbb N$, then $K$ is homeomorphic to $S$. In other words, we obtain a vector-valued Banach–Stone theorem which is an extension of a Gordon theorem and at the same time an improvement of a Behrends and Cambern theorem. In order to prove this, we show that if there exists an embedding $T$ of a $C(K)$ space into a $C(S, X)$ space, with distortion strictly less than $3$, then the cardinality of the $\alpha$th derivative of $S$ is finite or greater than or equal to the cardinality of the $\alpha$th derivative of $K$, for every ordinal $\alpha$.
