An ordinal LP-index for Banach spaces, with application to complemented subspaces of LP

@article{Bourgain1981AnOL,
  title={An ordinal LP-index for Banach spaces, with application to complemented subspaces of LP},
  author={Jean Bourgain and Haskell P. Rosenthal and Gideon Schechtman},
  journal={Annals of Mathematics},
  year={1981},
  volume={114},
  pages={193}
}
One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five "simple" examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These… Expand
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References

SHOWING 1-10 OF 17 REFERENCES
The ℒp spaces
The ℒp spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒp space if and only ifX* is and ℒq space (p−1+q−1=1). (ii) AExpand
EmbeddingL1 in a Banach lattice
We show that ifX is a Banach lattice containing no copy ofc0 and ifZ is a subspace ofX isomorphic toL1[0, 1] then (a)Z contains a subspaceZ0 isomorphic toL1 and complemented inX and (b)X contains aExpand
Examples of ℒp spaces (1
We present a simple method for constructing new ℒp spaces (1<p≠2<∞) out of old ones. Using this method and results of H.P. Rosenthal we prove the existence of a sequence of mutually nonisomorphicExpand
Classical Banach spaces
TLDR
Springer-Verlag is reissuing a selected few of these highly successful books in a new, inexpensive sofcover edition to make them easily accessible to younger generations of students and researchers. Expand
On subsequences of the Haar system inLp [0, 1], (1
We show that ifX is the closed linear span inLp[0,1] of a subsequence of the Haar system, thenX is isomorphic either tolp or toLp [0,1], [1<p<∞]. We give criteria to determine which of these casesExpand
Topics in Harmonic Analysis Related to the Littlewood-Paley Theory.
This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems,Expand
On subsequences of the Haar system inC(Δ)
Spaces arising as spans of subsequences of the Haar system inC(Δ) are studied. It is shown that for any compact metric spaceH there is a subsequence whose span is isomorphic toC(H), yet thatExpand
Les dérivations en théorie descriptive des ensembles et le théorème de la borne
© Springer-Verlag, Berlin Heidelberg New York, 1977, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avecExpand
Symmetric Structures in Banach Spaces
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