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The $\ell^{1}$-indices of Tsirelson type spaces

Authors: Denny H. Leung and Wee-Kee Tang
Journal: Proc. Amer. Math. Soc. 131 (2003), 511-521
MSC (2000): Primary 46B20; Secondary 05C05
Published electronically: June 3, 2002
MathSciNet review: 1933342
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Abstract: If $\alpha$ and $\beta$ are countable ordinals such that $\beta\neq0$, denote by $\overset{_{\sim}}{T}_{\alpha,\beta}$ the completion of $c_{00}$ with respect to the implicitly defined norm

\begin{displaymath}\Vert x\Vert=\max\{\Vert x\Vert_{\mathcal{S}_{\alpha}},\frac{1}{2}\sup \sum_{i=1}^{j}\Vert E_{i}x\Vert\}, \end{displaymath}

where the supremum is taken over all finite subsets $E_{1},\dots,E_{j}$ of $\mathbb{N} $ such that $E_{1}<\dots<E_{j}$ and $\{\min E_{1},\dots,\min E_{j}\}\in\mbox{$\mathcal{S}$ }_{\beta}$. It is shown that the Bourgain $\ell^{1}$-index of $\overset{_{\sim}}{T}_{\alpha,\beta} $is $\omega ^{\alpha+\beta\cdot\omega}$. In particular, if $\omega_{1}>\alpha =\omega^{\alpha_{1}}\cdot m_{1}+\dots+\omega^{\alpha_{n}}\cdot m_{n}$ in Cantor normal form and $\alpha_{n}$ is not a limit ordinal, then there exists a Banach space whose $\ell^{1}$-index is $\omega^{\alpha}$.

References [Enhancements On Off] (What's this?)

  • 1. D. E. Alspach and S. Argyros, Complexity of weakly null sequences, Diss. Math., 321 (1992), 1-44. MR 93j:46014
  • 2. S. A. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157-193. MR 99m:46021
  • 3. D. E. Alspach, R. Judd and E. Odell, The Szlenk index and local $\ell_{1}$-indices, preprint.
  • 4. J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Bel., 32 (1980), 235-249. MR 84e:46018
  • 5. P. G. Casazza, W. B. Johnson and L. Tzafriri, On Tsirelson's space, Israel J. Math. 47 (1984), 81-98. MR 85m:46013
  • 6. T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no $\ell_{p}$, Compositio Math. 29 (1974), 179-190. MR 50:8011
  • 7. I. Gasparis, A dichotomy theorem for subsets of the power set of the natural numbers, Proc. Amer. Math. Soc. 129 (2001), 759-764. MR 2001f:03089
  • 8. R. Judd, E. Odell, Concerning the Bourgain $\ell_{1}$ index of a Banach space, Israel J. Math. 108 (1998), 145-171. MR 2000k:46013
  • 9. P. Kiriakouli, Characterizations of spreading models of $\ell_{1} $, Comment. Math. Univ. Carolinae, 41(2000), 79-95. MR 2001e:46040
  • 10. D. H. Leung and W.-K. Tang, The Bourgain $\ell^{1}$-indices of mixed Tsirelson spaces, in preparation.
  • 11. Edward Odell, Nicole Tomczak-Jaegermann, and Roy Wagner, Proximity to $\ell_{1}$ and distortion in asymptotic $\ell_{1}$ spaces, Journal of Functional Analysis 150(1997), 101-145. MR 2000c:46027
  • 12. B. S. Tsirelson, Not every Banach space contains an embedding of $\ell_{p}$ or $c_{0}$, Functional Anal. Appl. 8 (1974), 138-141.

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Additional Information

Denny H. Leung
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543

Wee-Kee Tang
Affiliation: Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

Received by editor(s): March 7, 2001
Received by editor(s) in revised form: July 10, 2001, and September 20, 2001
Published electronically: June 3, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society