On dual spaces with bounded sequences without weak*-convergent convex blocks

Schlumprecht, Thomas

doi:10.1090/S0002-9939-1989-0979052-1

On dual spaces with bounded sequences without weak $\sp *$ -convergent convex blocks

Author: Thomas Schlumprecht
Journal: Proc. Amer. Math. Soc. 107 (1989), 395-408
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1989-0979052-1
MathSciNet review: 979052
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Abstract: In this work we show that if ${X^ * }$ contains bounded sequences without weak* convergent convex blocks, then it contains an isometric copy of ${L_1}\left( {{{\left\{ {0,1} \right\}}^{{\omega _1}}}} \right)$ .

[ABZ] S. A. Argyros, J. Bourgain and T. Zachariades, A result on the isomorphic embeddability of ${l_1}\left( \Gamma \right)$ , Stud. Math. 78 (1984), 77-92.
[B] J. Bourgain, La propiété de Radon-Nikodym, Publication des Math. de'l Université Pierre et Marie Currie, 36 (1979).
[BD] Jean Bourgain and Joe Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55–58. MR 774176, https://doi.org/10.1002/mana.19841190105
[DE] L. Drewnowski and G. Emmanuele, On Banach spaces with the Gel′fand-Phillips property. II, Rend. Circ. Mat. Palermo (2) 38 (1989), no. 3, 377–391 (1990). MR 1053378, https://doi.org/10.1007/BF02850021
[F] D. Fremlin, Consequences of Martin's axiom, Cambridge University Press, 1985.
[HJ] J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28 (1977), no. 4, 325–330. MR 482086, https://doi.org/10.1007/BF02760638
[Hy] Richard Haydon, An unconditional result about Grothendieck spaces, Proc. Amer. Math. Soc. 100 (1987), no. 3, 511–516. MR 891155, https://doi.org/10.1090/S0002-9939-1987-0891155-7
[HLO] Richard Haydon, Mireille Levy, and Edward Odell, On sequences without weak* convergent convex block subsequences, Proc. Amer. Math. Soc. 100 (1987), no. 1, 94–98. MR 883407, https://doi.org/10.1090/S0002-9939-1987-0883407-1
[O] E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 379–404. MR 606226
[P] A. Pełczyński, On Banach spaces containing 𝐿₁(𝜇), Studia Math. 30 (1968), 231–246. MR 232195, https://doi.org/10.4064/sm-30-2-231-246
[R] H. P. Rosenthal, On the structure of weak*-compact subsets of a dual Banach space, in preparation.
[S] Th. Schlumprecht, Limited sets in Banach spaces, Dissertation, München, 1987.
[T] Michel Talagrand, Un nouveau \cal𝐶(𝐾) qui possède la propriété de Grothendieck, Israel J. Math. 37 (1980), no. 1-2, 181–191 (French, with English summary). MR 599313, https://doi.org/10.1007/BF02762879