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Counterexamples to several problems concerning $ G\sb{\delta }$-embeddings

Authors: N. Ghoussoub and B. Maurey
Journal: Proc. Amer. Math. Soc. 92 (1984), 409-412
MSC: Primary 46B20; Secondary 54E52
MathSciNet review: 759665
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Abstract: Let JT (resp. JH) be the James-tree space (resp. the Hagler space). We observe that two canonical subspaces of $ {\text{J}}{{\text{T}}^ * }$ and $ {\text{J}}{{\text{H}}^ * }$, $ {G_\delta }$-embed in $ {l_2}$ even though they fail the Radon-Nikodym property. Such $ {G_\delta }$-embeddings cannot be the product of a finite number of semi-embeddings. This answers negatively several questions of Bourgain-Rosenthal [1] and Rosenthal [8].

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  • [1] J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), 149-188. MR 707202 (85g:46018)
  • [2] G. A. Edgar and R. F. Wheeler, Topological properties of Banach spaces, Pacific J. Math. (to appear). MR 765190 (86e:46013)
  • [3] N. Ghoussoub and H. P. Rosenthal, Martingales, $ {G_\delta }$-embeddings and quotients of $ {L_1}$, Math. Ann. 264 (1983), 321-332. MR 714107 (85e:46013)
  • [4] N. Ghoussoub and B. Maurey, $ {G_\delta }$-embeddings in Hilbert space, I, J. Funct. Anal. (to appear). MR 943500 (90b:46033)
  • [5] J. Hagler, A counterexample to several questions about Banach spaces, Studia Math. 60 (1977), 289-308. MR 0442651 (56:1032)
  • [6] R. C James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743. MR 0417763 (54:5811)
  • [7] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $ {l_1}$ and whose duals are not separable, Studia Math. 54 (1975), 81-105. MR 0390720 (52:11543)
  • [8] H. P. Rosenthal, Geometric properties related to the Radon-Nikodym property. Séminaire Choquet-Rogalski-St. Raymond, 20 année, 1980-1981, p. 20. MR 670804 (83k:46026)

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