Embeddings of $C(K)$ spaces into $C(S, X)$ spaces with distortion strictly less than 3

Leandro Candido ¹   ; Elói Medina Galego ¹

¹ Department of Mathematics, IME University of São Paulo Rua do Matão 1010 São Paulo, Brazil

Fundamenta Mathematicae, Tome 220 (2013) no. 1, pp. 83-92

Résumé

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 $\alpha$.

Détail

DOI : 10.4064/fm220-1-5

Keywords: spirit classical banach stone theorem prove intervals ordinals banach space having non trivial cotype existence isomorphism distortion strictly implies finite topological sum homeomorphic finite topological sum moreover contains subspace isomorphic every mathbb homeomorphic other words obtain vector valued banach stone theorem which extension gordon theorem time improvement behrends cambern theorem order prove there exists embedding space space distortion strictly cardinality alphath derivative finite greater equal cardinality alphath derivative every ordinal nbsp alpha