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Examples concerning heredity problems of WCG Banach spaces


Authors: Spiros A. Argyros and Sophocles Mercourakis
Journal: Proc. Amer. Math. Soc. 133 (2005), 773-785
MSC (2000): Primary 46B20, 46B26, 03E05
DOI: https://doi.org/10.1090/S0002-9939-04-07532-X
Published electronically: August 20, 2004
MathSciNet review: 2113927
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Abstract: We present two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space $X$ such that $X^{**}$ is WCG and $X^{**}$ does not contain $\ell^1$. The non-WCG subspace $Y$ of $X$ has the additional property that $Y^{**}$ is not WCG and $X/Y$ is reflexive.


References [Enhancements On Off] (What's this?)

  • [A] S.A. Argyros, Weakly Lindelöf Determined Banach spaces not containing $\ell^1(\mathbb{N} )$, (1993), preprint.
  • [ACGJM] S.A. Argyros, J.F. Castillo, A.S. Granero, M. Jimenez and J.P. Moreno, Complementation and embeddings of $c_0(I)$ in Banach spaces, Proc. London Math. Soc. (3) 85, (2002), 742-768. MR 2003k:46022
  • [AL] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Annals of Math., 88, (1968), 35-46. MR 37:4562
  • [AM] S.A. Argyros and S. Mercourakis, On weakly Lindelöf Banach spaces, Rocky Mountain J. Math. 23, (1993), 395-446. MR 94i:46016
  • [F] M.J. Fabián, Gâteaux Differentiability of Convex Functions and Topology, John Wiley & Sons, Inc. (1997). MR 98h:46009
  • [F1] M.J. Fabián, Each weakly countably determined Asplund space admits a Fréchet differentiable norm, Bull. Australian Math. Soc., 36, (1987), 367-374. MR 89c:46018
  • [H] R.G. Haydon, Some more characterizations of Banach spaces containing $\ell^1$, Math. Proc. Cambridge Phil. Soc., 80, (1976), 269-276. MR 54:11031
  • [HHZ] P. Habala, P. Hajek, and V. Zizler, Introduction to Banach Spaces I, II Matfyzpress, Prague (1996).
  • [HO] J. Hagler and E. Odell, A Banach space not containing $\ell^1$ whose dual ball is not weak$^*$sequentially compact, Illinois J. Math. 22, (1978), 290-294. MR 58:2174
  • [J] R.C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80, (1974), 738-743. MR 54:5811
  • [JL] W.B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17, (1974), 219-230. MR 54:5808
  • [JZ] K. John and V. Zizler, Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Funct. Anal. 15, (1974), 161-166. MR 54:5807
  • [LS] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $\ell^1$ and whose duals are nonseparable, Studia Math. 54, (1975), 81-105. MR 52:11543
  • [MS] S. Mercourakis and E. Stamati, A new class of weakly $\mathcal{K}$-analytic Banach spaces, Mathematika (to appear).
  • [R] H.P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compos. Math. 28, (1974), 83-111. MR 54:5810
  • [T] M. Talagrand, Espaces de Banach faiblement $\mathcal{K}$-analytiques, Ann. of Math. 110, (1979), 407-438. MR 81a:46021
  • [Z] V. Zizler, Nonseparable Banach Spaces, Handbook of the Geometry of Banach spaces W.B. Johnson, J. Lindenstrauss (eds.), North-Holland, Vol. 2, ch. 41, (2003), 1743-1816.

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Additional Information

Spiros A. Argyros
Affiliation: Department of Mathematics, National Technical University of Athens, Athens 15780, Greece
Email: sargyros@math.ntua.gr

Sophocles Mercourakis
Affiliation: Department of Mathematics, University of Athens, Athens 15784, Greece
Email: smercour@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9939-04-07532-X
Keywords: WCG Banach space, unconditional basis, tree
Received by editor(s): July 16, 2003
Received by editor(s) in revised form: October 23, 2003
Published electronically: August 20, 2004
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society

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