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A Classification of Tsirelson Type Spaces

Published online by Cambridge University Press:  20 November 2018

J. Lopez-Abad  and
A. Manoussakis
J. Lopez-Abad
Affiliation:
Equipe de Logique Mathématique, Université Paris Diderot-Paris 7, C.N.R.S.-UMR 7056, 75251 Paris Cedex 05, France e-mail:abad@logique.jussieu.fr
A. Manoussakis
Affiliation:
Department of Mathematics, University of Aegean, Karlovasi, Samos, GR 83200, Greece e-mail:amanouss@aegean.gr
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Abstract

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We give a complete classification of mixed Tsirelson spaces $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ for finitely many pairs of given compact and hereditary families ${{F}_{i}}$ of finite sets of integers and $0<{{\theta }_{i}}<1$ in terms of the Cantor–Bendixson indices of the families ${{F}_{i}}$, and ${{\theta }_{i}}(1\le i\le r)$. We prove that there are unique countable ordinal $\alpha $ and $0<\theta <1$ such that every block sequence of $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ has a subsequence equivalent to a subsequence of the natural basis of the $T({{S}_{{{\omega }^{\alpha }}}},\theta )$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.

Keywords

46B2005D10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 1108 - 1148
Copyright
Copyright © Canadian Mathematical Society 2008

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