On the geometry of higher order Schreier spaces

Abstract

For each countable ordinal α, let ${\mathcal{S}}_{\mathit{\alpha }}$ be the Schreier set of order α and $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ be the corresponding Schreier space of order α. In this paper, we prove several new properties of these spaces.

• 1. If α is nonzero, then $X_{{\mathcal{S}}_{\mathit{\alpha }}}$possesses the λ-property of Aron and Lohman and is a $(V)$-polyhedral space in the sense of Fonf and Vesely.

• 2. If α is nonzero and $1<p<\mathrm{\infty }$, then the p-convexification $X_{{\mathcal{S}}_{\mathit{\alpha }}}^p$ possesses the uniform λ-property of Aron and Lohman.

• 3. For each countable ordinal α, the space $X_{{\mathcal{S}}_{\mathit{\alpha }}}^{\ast }$ has the λ-property.

• 4. For $n\in \mathbb{N}$, if $T:X_{{\mathcal{S}}_n}\to X_{{\mathcal{S}}_n}$is an onto linear isometry, then $T{e}_i=\pm e_i$ for each $i\in \mathbb{N}$. Consequently, these spaces are light in the sense of Megrelishvili.

The fact that for nonzero α, $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ is polyhedral and has the λ-property implies that each $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ is an example of a space that solves a problem of Lindenstrauss from 1966 about the existence of a polyhedral infinite dimensional Banach space whose unit ball is the closed convex hull of its extreme points. The first example of such a space was given by De Bernardi in 2017 using a renorming of $c_0$.