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

Leandro Candido, and Elói Medina Galego. "Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3." Fundamenta Mathematicae 220.1 (2013): 83-92. <http://eudml.org/doc/286551>.

@article{LeandroCandido2013,
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 Xⁿ contains no subspace isomorphic to $X^\{n+1\}$ 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.},
author = {Leandro Candido, Elói Medina Galego},
journal = {Fundamenta Mathematicae},
keywords = {isomorphisms; spaces of vector-valued continuous functions; Banach-Stone theorem},
language = {eng},
number = {1},
pages = {83-92},
title = {Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3},
url = {http://eudml.org/doc/286551},
volume = {220},
year = {2013},
}

TY - JOUR
AU - Leandro Candido
AU - Elói Medina Galego
TI - Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3
JO - Fundamenta Mathematicae
PY - 2013
VL - 220
IS - 1
SP - 83
EP - 92
AB - 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 Xⁿ contains no subspace isomorphic to $X^{n+1}$ 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.
LA - eng
KW - isomorphisms; spaces of vector-valued continuous functions; Banach-Stone theorem
UR - http://eudml.org/doc/286551
ER -