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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 $\beta \neq 0$, denote by $tilde{T}_{\alpha ,\beta }$ the completion of $c_{00}$ with respect to the implicitly defined norm \[ \Vert x\Vert = \max \{\Vert x\Vert _{\mathcal {S}_{\alpha }},\frac {1}{2}\sup \sum _{i=1}^{j}\Vert E_{i}x\Vert \}, \] where the supremum is taken over all finite subsets $E_{1}$, โ€ฆ, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{๐›ฝ}$. It is shown that the Bourgain$โ„“^{1}$-index of$Tฬƒ_{๐›ผ,๐›ฝ}$is$๐œ”^{๐›ผ+๐›ฝโ‹…๐œ”}$. In particular, if$๐œ”_{1}>๐›ผ=๐œ”^{๐›ผ_{1}}โ‹…m_{1}+โ€ฆ+๐œ”^{๐›ผ_{n}}โ‹…m_{n}$in Cantor normal form and$๐›ผ_{n}$is not a limit ordinal, then there exists a Banach space whose$โ„“^{1}$-index is$๐œ”^{๐›ผ}$.$

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Additional Information

Denny H. Leung
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543
MR Author ID: 113100

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