More Quasi-Reflexive Subspaces

Abstract

It is shown that nonreflexive Banach spaces with a separable dual and the boundedly complete skipped blocking property have quasi-reflexive subspaces. In particular, Bourgain's somewhat reflexive ?,c-spaces and Polish Banach spaces are somewhat quasi-reflexive. The author in [1] has shown that most collections of Banach spaces which are known to contain a reflexive subspace also contain a quasi-reflexive subspace. (Of course, the words are needed to make this correct. Banach spaces are assumed to be infinite dimensional unless otherwise stated.) Indeed, the following question is still open: (1) Does each nonreflexive Banach space contain co, 11 or a quasi-reflexive subspace? An affirmative answer to (1) w6uld imply an affirmative answer to the well-known question: (2) Does each Banach space contain c0, 11 or a reflexive subspace? In [3] (or see [2]), Bourgain and Delbaen construct a collection of somewhat reflexive C,,-spaces. It was suggested to the author that these spaces might yield a negative answer to question (1). This note shows that these ZOO-spaces contain quasi-reflexive spaces, and hence (1) is still open. In [4] (or see [7]), Edgar and Wheeler show that each Polish Banach space contains a reflexive subspace. (A Banach space is Polish if its unit ball with the weak topology is a Polish topological space, i.e. homeomorphic to a separable complete metric space). Our Theorem 3 implies that each nonreflexive Polish space has quasireflexive subspaces. Indeed, X being Polish is equivalent to X* being separable and X has PCP [4] (or see [7]); while X has PCP is equivalent to X having the boundedly complete skipped blocking property [5] (or see [7]). We mention a short footnote to [1]. In [1] it was shown that most positive answers to question (2) also have a positive answer to question (1). The only exception noted in [1] was the space X* when X**/X is separable. However, Valdivia [8] had already shown that X = R Ef Y where Y** is separable and R is reflexive. Hence X* = R* E3 Y* and X* has quasi-reflexive subspaces by Theorem 9 of [1]. Our notation is standard and follows that of [6]. Also this paper is a sequel to [1], where some details are more carefully explained. Received by the editors April 1, 1986 and, in revised form, September 2, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B10, 46B20.