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 strictly less than 3 implies that some finite topological sum of K is homeomorphic to some finite topological sum of S. Moreover, if Xⁿ contains no subspace isomorphic to for every n ∈ 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 αth derivative of S is finite or greater than or equal to the cardinality of the αth derivative of K, for every ordinal α.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-1-5,
author = {Leandro Candido and El\'oi Medina Galego},
title = {Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {83-92},
zbl = {1271.46005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-1-5}
}
Leandro Candido; Elói Medina Galego. Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3. Fundamenta Mathematicae, Tome 220 (2013) pp. 83-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-1-5/