Interpolating hereditarily indecomposable Banach spaces

Argyros, S. A.; Felouzis, V.

doi:10.1090/S0894-0347-00-00325-8

Interpolating hereditarily indecomposable Banach spaces
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by S. A. Argyros and V. Felouzis;

J. Amer. Math. Soc. 13 (2000), 243-294

DOI: https://doi.org/10.1090/S0894-0347-00-00325-8

Published electronically: January 31, 2000

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Abstract:

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

Dale E. Alspach and Spiros Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44. MR 1191024
S. A. Argyros and I. Deliyanni. Banach spaces of the type of Tsirelson, preprint 1992.
S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. MR 1390965, DOI 10.1090/S0002-9947-97-01774-1
S. A. Argyros, I. Deliyanni, D. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. of Func. Anal. 159 (1998), 43-109.
S. A. Argyros and I. Gasparis. Unconditional structures of weakly null sequences (Preprint).
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, DOI 10.1007/BF02764008
G. Androulakis and E. Odell. Distorting mixed Tsirelson spaces (Preprint).
Steven F. Bellenot, Tsirelson superspaces and $l_p$, J. Funct. Anal. 69 (1986), no. 2, 207–228. MR 865221, DOI 10.1016/0022-1236(86)90089-3
J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belg. Sér. B 32 (1980), no. 2, 235–249. MR 682645
J. Bourgain. La propriété de Radon-Nicodym, Math. Univ. Pierre et Marie Curie 36 (1979).
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 355536, DOI 10.1016/0022-1236(74)90044-5
V. Ferenczi. Quotient Hereditarily Idecomposable spaces (Preprint).
V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225. MR 1489106, DOI 10.1007/BF02773800
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
W. T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (1996), no. 6, 1083–1093. MR 1421876, DOI 10.1007/BF02246998
W. T. Gowers, A Banach space not containing $c_0,\ l_1$ or a reflexive subspace, Trans. Amer. Math. Soc. 344 (1994), no. 1, 407–420. MR 1250820, DOI 10.1090/S0002-9947-1994-1250820-0
W. Timothy Gowers, A remark about the scalar-plus-compact problem, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 111–115. MR 1665582
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
P. Habala. Banach spaces all of whose subspaces fail the Gordon-Lewis property, Math. Ann. 310 (1998), N$^{\text {o}}$2, 197-212.
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
K. Kuratowski and A. Mostowski, Set theory, PWN—Polish Scientific Publishers, Warsaw; North-Holland Publishing Co., Amsterdam, 1968. Translated from the Polish by M. Maczyński. MR 229526
Nigel Kalton and Albert Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc. 57 (1976), no. 2, 251–255. MR 473896, DOI 10.1090/S0002-9939-1976-0473896-1
H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 493279
Denny H. Leung, On $c_0$-saturated Banach spaces, Illinois J. Math. 39 (1995), no. 1, 15–29. MR 1299646
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 500056
B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), no. 1, 77–98. MR 438091, DOI 10.4064/sm-61-1-77-98
Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
R. Neidinger. Factoring Operators through hereditarily-$\ell _p$ spaces, Lecture Notes in Math. 1166 (1985).
R. Neidinger. Properties of Tauberian Operators, Dissertation. University of Texas at Austin, 1984.
Richard Neidinger and Haskell P. Rosenthal, Norm-attainment of linear functionals on subspaces and characterizations of Tauberian operators, Pacific J. Math. 118 (1985), no. 1, 215–228. MR 783025
Edward Odell and Thomas Schlumprecht, The distortion problem, Acta Math. 173 (1994), no. 2, 259–281. MR 1301394, DOI 10.1007/BF02398436
E. Odell and T. Schlumprecht. A Banach space block finitely universal for monotone bases, Trans. Amer. Math. Soc. (to appear).
Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845
N. Tomczak-Jaegermann, Banach spaces of type $p$ have arbitrarily distortable subspaces, Geom. Funct. Anal. 6 (1996), no. 6, 1074–1082. MR 1421875, DOI 10.1007/BF02246997
A. Tsarpalias, A note on the Ramsey property, Proc. Amer. Math. Soc. 127 (1999), no. 2, 583–587. MR 1458267, DOI 10.1090/S0002-9939-99-04518-9
B. S. Tsirelson. Not every Banach space contains $\ell _p$ or $c_0$, Funct. Anal. Appl. 8 (1974), 138-141.