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

DOI:
https://doi.org/10.1090/S0002-9939-02-06586-3

Published electronically:
June 3, 2002

MathSciNet review:
1933342

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Abstract | References | Similar Articles | Additional Information

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 $E_{1}$, โฆ, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{๐ฝ}$. It is shown that the Bourgain$โ^{1}$-index of$Tฬ_{๐ผ,๐ฝ}$is$๐^{๐ผ+๐ฝโ ๐}$. In particular, if$๐_{1}>๐ผ=๐^{๐ผ_{1}}โ m_{1}+โฆ+๐^{๐ผ_{n}}โ m_{n}$in Cantor normal form and$๐ผ_{n}$is not a limit ordinal, then there exists a Banach space whose$โ^{1}$-index is$๐^{๐ผ}$.$

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Additional Information

**Denny H. Leung**

Affiliation:
Department of Mathematics, National University of Singapore, Singapore 117543

MR Author ID:
113100

Email:
matlhh@nus.edu.sg

**Wee-Kee Tang**

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

Email:
wktang@nie.edu.sg

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