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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

Weakly null sequences in $ L_1$


Authors: William B. Johnson, Bernard Maurey and Gideon Schechtman
Journal: J. Amer. Math. Soc. 20 (2007), 25-36
MSC (2000): Primary 46B15, 46E30
Published electronically: September 19, 2006
MathSciNet review: 2257395
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Abstract: We construct a weakly null normalized sequence $ \{f_i\}_{i=1}^{\infty}$ in $ L_1$ so that for each $ \varepsilon>0$, the Haar basis is $ (1+\varepsilon)$-equivalent to a block basis of every subsequence of $ \{f_i\}_{i=1}^{\infty}$. In particular, the sequence $ \{f_i\}_{i=1}^{\infty}$ has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces.


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Additional Information

William B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: johnson@math.tamu.edu

Bernard Maurey
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université de Marne la Vallée, 77454 Champs-sur-Marne, France
Email: maurey@univ-mlv.fr

Gideon Schechtman
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Email: gideon.schechtman@weizmann.ac.il

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00548-0
PII: S 0894-0347(06)00548-0
Keywords: $L_1$, Haar basis, unconditional basic sequence, weakly null.
Received by editor(s): June 8, 2004
Published electronically: September 19, 2006
Additional Notes: The first author was supported in part by NSF grant DMS-0200690 and NSF grant DMS-0503688, Texas Advanced Research Program 010366-0033-20013 and the U.S.-Israel Binational Science Foundation.
The last author was supported in part by the Israel Science Foundation and the U.S.-Israel Binational Science Foundation and was a participant in the NSF Workshop in Linear Analysis and Probability, Texas A&M University.
Article copyright: © Copyright 2006 by the authors