The -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If and are countable ordinals such that , denote by the completion of with respect to the implicitly defined norm

where the supremum is taken over all finite subsets of such that and . It is shown that the Bourgain -index of is . In particular, if in Cantor normal form and is not a limit ordinal, then there exists a Banach space whose -index is .

**1.**Dale E. Alspach and Spiros Argyros,*Complexity of weakly null sequences*, Dissertationes Math. (Rozprawy Mat.)**321**(1992), 44. MR**1191024****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**1658551**, 10.1007/BF02764008**3.**D. E. Alspach, R. Judd and E. Odell, The Szlenk index and local -indices, preprint.**4.**J. Bourgain,*On convergent sequences of continuous functions*, Bull. Soc. Math. Belg. Sér. B**32**(1980), no. 2, 235–249. MR**682645****5.**P. G. Casazza, W. B. Johnson, and L. Tzafriri,*On Tsirelson’s space*, Israel J. Math.**47**(1984), no. 2-3, 81–98. MR**738160**, 10.1007/BF02760508**6.**T. Figiel and W. B. Johnson,*A uniformly convex Banach space which contains no 𝑙_{𝑝}*, Compositio Math.**29**(1974), 179–190. MR**0355537****7.**I. Gasparis,*A dichotomy theorem for subsets of the power set of the natural numbers*, Proc. Amer. Math. Soc.**129**(2001), no. 3, 759–764. MR**1707146**, 10.1090/S0002-9939-00-05594-5**8.**Robert Judd and Edward Odell,*Concerning Bourgain’s 𝑙₁-index of a Banach space*, Israel J. Math.**108**(1998), 145–171. MR**1669388**, 10.1007/BF02783046**9.**P. Kiriakouli,*Characterizations of spreading models of 𝑙¹*, Comment. Math. Univ. Carolin.**41**(2000), no. 1, 79–95. MR**1756928****10.**D. H. Leung and W.-K. Tang, The Bourgain -indices of mixed Tsirelson spaces, in preparation.**11.**Edward Odell, Nicole Tomczak-Jaegermann, and Roy Wagner,*Proximity to 𝑙₁ and distortion in asymptotic 𝑙₁ spaces*, J. Funct. Anal.**150**(1997), no. 1, 101–145. MR**1473628**, 10.1006/jfan.1997.3106**12.**B. S. Tsirelson, Not every Banach space contains an embedding of or , Functional Anal. Appl. 8 (1974), 138-141.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46B20,
05C05

Retrieve articles in all journals with MSC (2000): 46B20, 05C05

Additional Information

**Denny H. Leung**

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

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

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

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