Uniqueness of unconditional bases in c 0 -products

P. Casazza; N. Kalton

Studia Mathematica (1999)

  • Volume: 133, Issue: 3, page 275-294
  • ISSN: 0039-3223

Abstract

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We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does c 0 ( X ) . We also give some positive results including a simpler proof that c 0 ( ℓ 1 ) has a unique unconditional basis and a proof that c 0 ( ℓ p n N n ) has a unique unconditional basis when p n ￬ 1 , N n + 1 ≥ 2 N n and ( p n - p n + 1 ) l o g N n remains bounded.

How to cite

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Casazza, P., and Kalton, N.. "Uniqueness of unconditional bases in $c_0$-products." Studia Mathematica 133.3 (1999): 275-294. <http://eudml.org/doc/216619>.

@article{Casazza1999,
abstract = {We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0(X)$. We also give some positive results including a simpler proof that $c_0(ℓ_1)$ has a unique unconditional basis and a proof that $c_0(ℓ_\{p_n\}^\{N_n\})$ has a unique unconditional basis when $p_n ￬ 1$, $N_\{n+1\} ≥ 2N_\{n\}$ and $(p_n-p_\{n+1\}) logN_\{n\}$ remains bounded.},
author = {Casazza, P., Kalton, N.},
journal = {Studia Mathematica},
keywords = {space ; unconditional basis; Tsirelson space},
language = {eng},
number = {3},
pages = {275-294},
title = {Uniqueness of unconditional bases in $c_0$-products},
url = {http://eudml.org/doc/216619},
volume = {133},
year = {1999},
}

TY - JOUR
AU - Casazza, P.
AU - Kalton, N.
TI - Uniqueness of unconditional bases in $c_0$-products
JO - Studia Mathematica
PY - 1999
VL - 133
IS - 3
SP - 275
EP - 294
AB - We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0(X)$. We also give some positive results including a simpler proof that $c_0(ℓ_1)$ has a unique unconditional basis and a proof that $c_0(ℓ_{p_n}^{N_n})$ has a unique unconditional basis when $p_n ￬ 1$, $N_{n+1} ≥ 2N_{n}$ and $(p_n-p_{n+1}) logN_{n}$ remains bounded.
LA - eng
KW - space ; unconditional basis; Tsirelson space
UR - http://eudml.org/doc/216619
ER -

References

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  1. [1] B. Bollobas, Combinatorics, Cambridge Univ. Press, 1986. 
  2. [2] J. Bourgain, On the Dunford-Pettis property, Proc. Amer. Math. Soc. 81 (1981), 265-272. Zbl0463.46027
  3. [3] J. Bourgain, P. G. Casazza, J. Lindenstrauss and L. Tzafriri, Banach spaces with a unique unconditional basis, up to a permutation, Mem. Amer. Math. Soc. 322 (1985). Zbl0575.46011
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  10. [10] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in L p -spaces and their applications, Studia Math. 29 (1968), 315-349. Zbl0183.40501
  11. [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, 1977. Zbl0362.46013
  12. [12] J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Funct. Anal. 3 (1969), 115-125. Zbl0174.17201
  13. [13] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Spaces, Lecture Notes in Math. 1200, Springer, 1986. Zbl0606.46013
  14. [14] B. S. Mityagin, Equivalence of bases in Hilbert scales, Studia Math. 37 (1970), 111-137 (in Russian). 
  15. [15] G. Pisier, The Volume of Convex Bodies and Geometry of Banach Spaces, Cambridge Tracts in Math. 94, Cambridge Univ. Press, 1989. 
  16. [16] P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, II, Israel J. Math 97 (1997), 253-280. Zbl0874.46007
  17. [17] M. Wojtowicz, On Cantor-Bernstein type theorems in Riesz spaces, Indag. Math. 91 (1988), 93-100. Zbl0654.46013

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