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The cofinal property of the reflexive indecomposable Banach spaces. (English. French summary) Zbl 1253.46009

It is shown that every separable reflexive Banach space is a quotient of a reflexive hereditarily indecomposable space. As a consequence, every separable reflexive Banach space is isomorphic to a subspace of a reflexive indecomposable space. Moreover, it is also proved that every separable reflexive Banach space is a quotient of a reflexive complementably \(\ell_{p}\)-saturated space with \(1<p< \infty\) or of a \(c_{0}\)-saturated space.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B06 Asymptotic theory of Banach spaces
46B70 Interpolation between normed linear spaces
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[1] Alspach, Dale E.; Argyros, Spiros A., Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.), 321 (1992) · Zbl 0787.46009
[2] Amir, D.; Lindenstrauss, J., The structure of weakly compact sets in Banach spaces, Ann. of Math. (2), 88, 35-46 (1968) · Zbl 0164.14903 · doi:10.2307/1970554
[3] Argyros, Spiros A.; Dodos, Pandelis, Genericity and amalgamation of classes of Banach spaces, Adv. Math., 209, 2, 666-748 (2007) · Zbl 1109.03047 · doi:10.1016/j.aim.2006.05.013
[4] Argyros, Spiros A.; Felouzis, V., Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc., 13, 2, 243-294 (electronic) (2000) · Zbl 0956.46014 · doi:10.1090/S0894-0347-00-00325-8
[5] Argyros, Spiros A.; Godefroy, Gilles; Rosenthal, Haskell P., Handbook of the geometry of Banach spaces, Vol. 2, 1007-1069 (2003) · Zbl 1121.46008 · doi:10.1016/S1874-5849(03)80030-X
[6] Argyros, Spiros A.; Haydon, Richard G., A hereditarily indecomposable \(\mathcal{ L}^\infty \)-space that solves the scalar-plus-compact problem, Acta Math., 206, 1, 1-54 (2011) · Zbl 1223.46007 · doi:10.1007/s11511-011-0058-y
[7] Argyros, Spiros A.; Mercourakis, S.; Tsarpalias, A., Convex unconditionality and summability of weakly null sequences, Israel J. Math., 107, 157-193 (1998) · Zbl 0942.46007 · doi:10.1007/BF02764008
[8] Argyros, Spiros A.; Todorcevic, Stevo, Ramsey methods in analysis (2005) · Zbl 1092.46002
[9] Argyros, Spiros A.; Tolias, Andreas, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc., 170, 806 (2004) · Zbl 1055.46004
[10] Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A., Factoring weakly compact operators, J. Functional Analysis, 17, 311-327 (1974) · Zbl 0306.46020 · doi:10.1016/0022-1236(74)90044-5
[11] Gasparis, I., New examples of \(c_0\)-saturated Banach spaces, Math. Ann., 344, 2, 491-500 (2009) · Zbl 1176.46017 · doi:10.1007/s00208-008-0319-z
[12] Gasparis, I., New examples of \(c_0\)-saturated Banach spaces. II, J. Funct. Anal., 256, 11, 3830-3840 (2009) · Zbl 1180.46007 · doi:10.1016/j.jfa.2008.11.021
[13] Gowers, W. T.; Maurey, B., The unconditional basic sequence problem, J. Amer. Math. Soc., 6, 4, 851-874 (1993) · Zbl 0827.46008 · doi:10.2307/2152743
[14] Grothendieck, A., Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math., 74, 168-186 (1952) · Zbl 0046.11702 · doi:10.2307/2372076
[15] Leung, Denny H., Some stability properties of \(c_0\)-saturated spaces, Math. Proc. Cambridge Philos. Soc., 118, 2, 287-301 (1995) · Zbl 0840.46010 · doi:10.1017/S0305004100073643
[16] Lindenstrauss, J., On non separable reflexive Banach spaces, Bull. Amer. Math. Soc., 72, 967-970 (1966) · Zbl 0156.36403 · doi:10.1090/S0002-9904-1966-11606-3
[17] Lindenstrauss, J., Some open problems in Banach space theory, Sémin. Choquet, 15e Année 1975/76, Initiation à l’Analyse, Exposé 18, 9 p. (1977) (1977) · Zbl 0363.46016
[18] Lopez-Abad, J.; Manoussakis, A., A classification of Tsirelson type spaces, Canad. J. Math., 60, 5, 1108-1148 (2008) · Zbl 1160.46008 · doi:10.4153/CJM-2008-049-0
[19] Neidinger, Richard D., Banach spaces (Columbia, Mo., 1984), 1166, 116-128 (1985) · Zbl 0587.47023 · doi:10.1007/BFb0074701
[20] Neidinger, Richard Dean, Properties of Tauberian Operators on Banach Spaces (semi-embedding, factorization, functional) (1984)
[21] Rosenthal, Haskell P., A characterization of Banach spaces containing \(l^1\), Proc. Nat. Acad. Sci. U.S.A., 71, 2411-2413 (1974) · Zbl 0297.46013 · doi:10.1073/pnas.71.6.2411
[22] Sobczyk, Andrew, Projection of the space \((m)\) on its subspace \((c_0)\), Bull. Amer. Math. Soc., 47, 938-947 (1941) · Zbl 0027.40801 · doi:10.1090/S0002-9904-1941-07593-2
[23] Zippin, M., Banach spaces with separable duals, Trans. Amer. Math. Soc., 310, 1, 371-379 (1988) · Zbl 0706.46015 · doi:10.2307/2001128
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