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April 2021 On the geometry of higher order Schreier spaces
Leandro Antunes, Kevin Beanland, Hùng Việt Chu
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Illinois J. Math. 65(1): 47-69 (April 2021). DOI: 10.1215/00192082-8827623

Abstract

For each countable ordinal α, let Sα be the Schreier set of order α and XSα 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 XSαpossesses 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<, then the p-convexification XSαp possesses the uniform λ-property of Aron and Lohman.

  • 3. For each countable ordinal α, the space XSα has the λ-property.

  • 4. For nN, if T:XSnXSnis an onto linear isometry, then Tei=±ei for each iN. Consequently, these spaces are light in the sense of Megrelishvili.

The fact that for nonzero α, XSα is polyhedral and has the λ-property implies that each XSα 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 c0.

Citation

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Leandro Antunes. Kevin Beanland. Hùng Việt Chu. "On the geometry of higher order Schreier spaces." Illinois J. Math. 65 (1) 47 - 69, April 2021. https://doi.org/10.1215/00192082-8827623

Information

Received: 9 August 2019; Revised: 30 July 2020; Published: April 2021
First available in Project Euclid: 16 December 2020

Digital Object Identifier: 10.1215/00192082-8827623

Subjects:
Primary: 46B03

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

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Vol.65 • No. 1 • April 2021
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