Applications of the theory of semi-embeddings to Banach space theory

AbstractLet 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 L1X. 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