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Interpolating hereditarily indecomposable Banach spaces

Authors: S. A. Argyros and V. Felouzis
Journal: J. Amer. Math. Soc. 13 (2000), 243-294
MSC (2000): Primary 46B20, 46B70; Secondary 46B03, 52A07, 03E05
Published electronically: January 31, 2000
MathSciNet review: 1750954
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Abstract | References | Similar Articles | Additional Information


The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to $\ell^1$, or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of $L^p(\lambda), 1<p<\infty$, it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator $I: L^{\infty}(\lambda)\to L^1(\lambda)$ admits a factorization through a H.I. space. The same result holds for every strictly singular operator $T: \ell^p\to \ell^q, 1<p,q<\infty$.

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.

References [Enhancements On Off] (What's this?)

  • [AA] D. Alspach and S. A. Argyros. Complexity of weakly null sequences, Dissertationes Mathematicae CCCXXI (1992). MR 93j:46014
  • [AD1] S. A. Argyros and I. Deliyanni. Banach spaces of the type of Tsirelson, preprint 1992.
  • [AD2] S. A. Argyros and I. Deliyanni. Examples of asymptotic $\ell _1$ Banach spaces, Trans. AMS 349 (1997), 973-995. MR 97f:46021
  • [ADKM] S. A. Argyros, I. Deliyanni, D. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. of Func. Anal. 159 (1998), 43-109. CMP 99:04
  • [AG] S. A. Argyros and I. Gasparis. Unconditional structures of weakly null sequences (Preprint).
  • [AMT] S. A. Argyros, S. Merkourakis and A. Tsarpalias. Convex unconditionality and summability of weakly null sequences, Isr. J. Math. 107 (1998), 157-193. MR 99m:46021
  • [AO] G. Androulakis and E. Odell. Distorting mixed Tsirelson spaces (Preprint).
  • [B] S. F. Bellenot. Tsirelson superspaces and $ \ell _p$, Journ. of Funct. Analysis. 69 (1986), No2, 207-228. MR 88f:46033
  • [Bo1] J. Bourgain. Convergent sequences of continuous functions, Bull. Soc. Math. Belg. Ser. B 32 (1980), 235-249. MR 84e:46018
  • [Bo2] J. Bourgain. La propriété de Radon-Nicodym, Math. Univ. Pierre et Marie Curie 36 (1979).
  • [D] J. Diestel. Sequences and series in Banach spaces, Graduate texts in Math. 92, Springer-Verlag, 1984. MR 85i:46020
  • [DFJP] W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski. Factoring Weakly Compact Operators, Journ. of Funct. Analysis 17 (1974), 311-327. MR 50:8010
  • [F1] V. Ferenczi. Quotient Hereditarily Idecomposable spaces (Preprint).
  • [F2] V. Ferenczi. A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199-225. MR 98m:46013
  • [J] R. C. James. Bases and reflexivity of Banach spaces, Ann. of Math. 52 (1950), 518-527. MR 12:616b
  • [G] W. T. Gowers. A new dichotomy for Banach spaces, GAFA 6 (1996), 1083-1093. MR 97m:46017
  • [G1] W. T. Gowers. A Banach space not containing $c_0$, $\ell _1$ or a reflexive subspace, Trans. Amer. Math. Soc. 344 (1994), 407-420. MR 94j:46024
  • [G2] W. T. Gowers. A remark about the Scalar-Plus-Compact Problem, Convex Geometric Analysis 34 (1998), 111-115. MR 99m:46015
  • [GM] W. T. Gowers and B. Maurey. The unconditional basic sequence problem, Journal of AMS 6 (1993), 851-874. MR 94k:46021
  • [Gr] A. Grothendieck. Critères de compacité dans les espaces fonctionelles généraux, Amer. J. Math. 74 (1952), 168-186. MR 13:857e
  • [H] P. Habala. Banach spaces all of whose subspaces fail the Gordon-Lewis property, Math. Ann. 310 (1998), N $^{\text{o}}$2, 197-212. CMP 98:07
  • [Ke] A. Kechris. Classical Descriptive Set Theory, Springer-Verlag, Berlin (1994). MR 96e:03057
  • [KM] K. Kuratowski and A. Mostowski. Set Theory, Amsterdam (1968). MR 37:5100
  • [KW] N. Kalton and A. Wilanski. Tauberian operators on Banach spaces, Proc. A.M.S. 57 (1976), 251-255. MR 57:13555
  • [L] H. E. Lacey. The Isometric Theory of Classical Banach Spaces, Springer-Verlag, Berlin, 1974. MR 58:12308
  • [Le] D. H. Leung. On $c_0$-saturated spaces, Illinois J. Math. 39 (1995), 15-29. MR 96h:46024
  • [LT] J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I, Springer Verlag 92, 1977. MR 58:17766
  • [MR] B. Maurey and H. Rosenthal. Normalized weakly null sequences with no unconditional subsequence , Studia Math. 61 (1977), 77-98. MR 55:11010
  • [MS] V. Milman and G. Schechtman. Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer-Verlag. MR 87m:46038
  • [N] R. Neidinger. Factoring Operators through hereditarily-$\ell _p$ spaces, Lecture Notes in Math. 1166 (1985).
  • [N1] R. Neidinger. Properties of Tauberian Operators, Dissertation. University of Texas at Austin, 1984.
  • [NR] R. Neidinger and H. P. Rosenthal. Norm-attainment of linear functional on subspaces and characterizations of Tauberian operators, Pac. J. Math. 118 (1985), 215-228. MR 86f:46013
  • [OS] E. Odell and T. Schlumprecht. The distortion problem, Acta Math. 173 (1994), 259-281. MR 96a:46031
  • [OS1] E. Odell and T. Schlumprecht. A Banach space block finitely universal for monotone bases, Trans. Amer. Math. Soc. (to appear). CMP 98:16
  • [S] T. Schlumprecht. An arbitrarily distortable Banach space, Israel J. Math 76 (1991), 81-95. MR 93h:46023
  • [To] N. Tomczak-Jaegermann. Banach spaces of type $p$ have arbitrarily distortable subspaces, GAFA 6 (1996), 1075-1082. MR 98g:46020
  • [T] A. Tsarpalias. A note on Ramsey property, Proc. A.M.S. 127 (1999), 583-587. MR 99c:04005
  • [Ts] B. S. Tsirelson. Not every Banach space contains $ \ell _p$ or $c_0$, Funct. Anal. Appl. 8 (1974), 138-141.

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Additional Information

S. A. Argyros
Affiliation: Department of Mathematics, University of Athens, Athens, Greece

V. Felouzis
Affiliation: Department of Mathematics, University of Athens, Athens, Greece

Keywords: Interpolation methods, hereditarily indecomposable spaces, thin convex sets, Schreier families, summability methods
Received by editor(s): April 14, 1998
Received by editor(s) in revised form: June 8, 1999
Published electronically: January 31, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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