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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 $\omega^{\alpha}$.

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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