Applications of the theory of semi-embeddings to Banach space theory |
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Authors: | J Bourgain HP Rosenthal |
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Institution: | Free University of Brussels, Brussels, Belgium;Department of Mathematics, University of Texas at Austin, Austin, Texas 78712 U.S.A. |
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Abstract: | Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L1 semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l1. As a consequence of the proof, it is shown that if X is a subspace of L1, either L1 embeds in X or l1 embeds in . The simpler result that L1 does not semi-embed in c0 is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed. |
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