An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

Abstract

Abstract. We create a new family of Banach spaces, the James-Schreier spaces, by amalgamating two important classical Banach spaces: James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. We then investigate the properties of these James-Schreier spaces, paying particular attention to how key properties of their 'ancestors' (that is, the James space and the Schreier space) are expressed in them. Our main results include that each James-Schreier space is c0-saturated and that no James-Schreier space embeds in a Banach space with an unconditional basis.1. Introduction. The purpose of this paper is to introduce a new family of Banach spaces which we call the James-Schreier spaces because they arise by amalgamating the definitions of the quasi-reflexive James spaces with the Schreier space. The original motivation behind these spaces was to produce a new example of a Banach sequence algebra with a bounded approximate identity, in analogy with Andrew and Green's study of the James space as a Banach algebra [3]. This idea turned out to be successful, as essentially all results about the James space as a Banach algebra carry over to our new spaces; see [9] for details.Having thus reached our initial goal, we soon realized that a serious problem was lurking in the background, namely: how can we distinguish the James-Schreier spaces from the James spaces? Obviously, if they were isomorphic, our findings would be of no interest. In order to resolve this problem, we turned to Banach-space properties and, as we shall see, at that level differences abound; this is the main theme of the present paper.

Some remarks on James–Schreier spaces

“…Before presenting these, we recall some notation and terminology from [5]. Throughout, K denotes the scalar field; either K = R or K = C. We write card A for the cardinality of a (typically finite) set A.…”

“…First, it was shown in [5] that (e n ) n∈N , where e n ∈ c 00 is the sequence with 1 in position n and 0 elsewhere, is a Schauder basis for V p for each p 1 and, moreover, that this basis is shrinking (meaning that the associated sequence of biorthogonal functionals (e n ) n∈N is a Schauder basis for the dual space V p ) whenever p > 1. The question of whether or not the basis (e n ) n∈N is shrinking for p = 1 was left open; in Section 2 we answer this question in the positive.…”

“…The Schreier spaces have recently been amalgamated with James' quasi-reflexive Banach spaces [6] by Bird and Laustsen [5]. More precisely, for 1 p < ∞, the p-th James-Schreier space, denoted V p , is the completion of c 00 with respect to the norm…”

“…,n k+1 ∈ N, k n 1 < n 2 < · · · < n k+1 , (1.2) where x = (α n ) n∈N ∈ c 00 . We refer to [5] for the background and motivation behind these spaces, as well as a thorough study of their fundamental properties. The purpose of this paper is to resolve two problems left open in [5].…”

“…As a consequence, we deduce that V 1 does not have Pełczyński's property (u) and hence does not embed in a Banach space with an unconditional Schauder basis. Second, regarding embeddings and isomorphisms of Schreier and James-Schreier spaces, it was proved in [5] that:…”