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Reflexivity and the separable quotient problem for a class of Banach spaces

Wójtowicz M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

383-394

Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of l1 iff for every n ≥ l, the nth dual X(n) of X contains no isomorphic copy of l1 iff X has no quotient isomorphic to c0 and X does not have a subspace isomorphic to l1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovskiĭ, Tzafriri, Meyer-Nieberg, and Diaz and Fernández. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c0, l1, or a reflexive space with a Schauder basis.