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2013 | 220 | 1 | 83-92
Tytuł artykułu

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

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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.
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  • Department of Mathematics, IME, University of São Paulo, Rua do Matão 1010, São Paulo, Brazil
  • Department of Mathematics, IME, University of São Paulo, Rua do Matão 1010, São Paulo, Brazil
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-1-5
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