Arbitrarily distortable Banach spaces of higher order

We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank AD (•), introduced by P. Dodos, uses the transfinite Schreier families and has the property that AD ( X ) < ω 1 if and only if X is arbitrarily distortabl...

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Bibliographic Details
Published in:Israel journal of mathematics 2016-07, Vol.214 (2), p.553-581
Main Authors: Beanland, Kevin, Causey, Ryan, Motakis, Pavlos
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Banach spaces
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank AD (•), introduced by P. Dodos, uses the transfinite Schreier families and has the property that AD ( X ) < ω 1 if and only if X is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order l 1 spreading models. We also compute this rank for several Banach spaces. In particular, it is shown that the class of Banach spaces , which each admit l 1 and c 0 spreading models hereditarily, and were introduced by S. A. Argyros, the first and third author, satisfy . This answers some questions of Dodos.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-016-1347-0