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Some Results on the Schroeder–Bernstein Property for Separable Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Valentin Ferenczi  and
Elόi Medina Galego
Valentin Ferenczi
Affiliation:
Institut de Mathématiques de Jussieu, Projet Analyse Fonctionnelle, Université Pierre et Marie Curie – Paris 6, Boîte 186, 4, Place Jussieu 75252, Paris Cedex 05, France e-mail: ferenczi@ccr.jussieu.fr
Elόi Medina Galego
Affiliation:
Department of Mathematics, IME, University of São Paulo, São Paulo 05315-970, Brazil e-mail: eloi@ime.usp.br, ferenczi@ime.usp.br
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Abstract

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We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is ${{2}^{{\aleph_{0}}}}$ . Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

Keywords

46B0346B20complemented subspacesSchroeder–Bernstein property

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 1 , 01 February 2007 , pp. 63 - 84
Copyright
Copyright © Canadian Mathematical Society 2007

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