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

  • Valentin Ferenczi (a1) and Elόi Medina Galego (a2)

Abstract

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.

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References

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[1] Casazza, P. G., Finite dimensional decompositions in Banach spaces. In: Geometry of Normed Linear Spaces. Contemp. Math. 52, American Mathematical Society, Providence, RI, 1986, pp. 129.
[2] Casazza, P. G., The Schroeder-Bernstein property for Banach space. In: Banach Space Theory. Contemp. Math. 85, American Mathematical Society, Providence, RI, 1989, pp. 6177.
[3] Castillo, J. M. F. and González, M., Three-Space Problems in Banach Spaces Theory. Lecture Notes in Mathematics 1667, Springer-Verlag, Berlin, 1997.
[4] Ferenczi, V., A uniformly convex and hereditarily indecomposable Banach space. Israel J. Math. 102(1997), 199225.
[5] Ferenczi, V. and Galego, E. M., Some equivalence relations which are Borel reducible to isomorphism between Banach spaces. Israel J. Math. 152(2006), 6182.
[6] Ferenczi, V. and Rosendal, C., On the number of non-isomorphic subspaces of a Banach space. Studia Math. 168(2005), no. 3, 203216.
[7] Ferenczi, V. and Rosendal, C., Ergodic Banach spaces. Adv. Math. 195(2005), no. 1, 259282.
[8] Galego, E. M., On solutions to the Schroeder-Bernstein problem for Banach spaces. Arch. Math. (Basel) 79(2002), no. 4, 299307.
[9] Galego, E. M., The Schroeder-Bernstein index for Banach spaces. Studia Math. 164(2004), no. 1, 2938.
[10] Gasparis, I., A continuum of totally incomparable hereditarily indecomposable Banach spaces. Studia Math. 151(2002), no. 3, 277298.
[11] Gowers, W. T., A solution to the Schroeder-Bernstein problem for Banach spaces. Bull. London Math. Soc. 28(1996), no. 3, 297304.
[12] Gowers, W. T. and Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc. 6(1993), no. 4, 851874.
[13] Gowers, W. T. and Maurey, B., Banach spaces with small spaces of operators. Math. Ann. 307(1997), no. 4, 543568.
[14] Herman, R. and Whitley, R., An example concerning reflexivity. Studia Math. 28(1966/67) 289294.
[15] Kalton, N. J., A remark on Banach spaces isomorphic to their squares. In: Function Spaces. Contemp. Math. 232, American Mathematical Society, Providence, RI, 1999, pp. 211217.
[16] Kechris, A. S.. Classical Descriptive Set Theory. Graduate Texts in Mathematics 156, Springer-Verlag, New York, 1995.
[17] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces. I. Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, New York. 1979.
[18] Rosendal, C., Incomparable, non isomorphic and minimal Banach spaces. Fund. Math. 183(2004), no. 3, 253274.
[19] Schlumprecht, T., An arbitrarily distortable Banach space. Israel J. Math. 76(1991), no. 1-2, 8195.
[20] Wojtaszczyk, P., On projections and unconditional bases in direct sums of Banach spaces. II. Studia Math. 62(1978), no. 2, 193201.
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Some Results on the Schroeder–Bernstein Property for Separable Banach Spaces

  • Valentin Ferenczi (a1) and Elόi Medina Galego (a2)

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