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Invariant subspaces of \(X^{**}\) under the action of biconjugates. (English) Zbl 1164.47302
Summary: We study conditions on an infinite-dimensional separable Banach space  \(X\) implying that \(X\)  is the only nontrivial invariant subspace of  \(X^{**}\) under the action of the algebra  \(\mathbb A(X)\) of biconjugates of bounded operators on  \(X\: \mathbb A(X)=\{T^{**}\: T \in \mathcal {B}(X)\}\). Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of  \(c_{0}\) and show in particular that any space which does not contain  \(\ell _1\) and has Pelczynski’s property (u) is simple.
MSC:
47A15 Invariant subspaces of linear operators
46B25 Classical Banach spaces in the general theory
47L05 Linear spaces of operators
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References:
[1] S. Argyros, G. Godefroy, and H. P. Rosenthal: Descriptive set theory and Banach spaces. Handbook of the geometry of Banach spaces, Vol. 2. North Holland, Amsterdam (2003), 1007–1069. · Zbl 1121.46008
[2] P. G. Casazza, R. H. Lohman: A general construction of spaces of the type of R. C. James. Canad. J. Math. 27 (1975), 1263–1270. · Zbl 0314.46017
[3] G. Godefroy, D. Li: Banach spaces which are M-ideals in their bidual have property (u). Ann. Inst. Fourier. 39 (1989), 361–371. · Zbl 0659.46014
[4] G. Godefroy, P. Saab: Weakly unconditionnaly converging series in M-ideals. Math. Scand. 64 (1989), 307–318. · Zbl 0676.46006
[5] G. Godefroy, M. Talagrand: Nouvelles classes d’espaces de Banach à prédual unique. Séminaire d’analyse fonctionnelle École Polytechnique, Exposé n o 9 (année 1980–1981).
[6] G. Godefroy: Existence and uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, IA, 1987). Contemp. Math. 85 (1989), 131–193.
[7] G. Godefroy, N. J. Kalton, and P. D. Saphar: Unconditional ideals in Banach spaces. Studia Math. 104 (1993), 13–59. · Zbl 0814.46012
[8] M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant, and V. Zizler: Functional Analysis and Infinite Dimensional Geometry. CMS books in Mathematics/Ouvrages de Mathématiques de la SMC, Vol. 8. Springer-Verlag, New York, 2001.
[9] R. Haydon, E. Odell, and H. P. Rosenthal: On certain classes of Baire-1 functions with applications to Banach space theory. Functional Analysis. Springer-Verlag, 1991, pp. 1–35. · Zbl 0762.46006
[10] R. C. James: Bases and reflexivity of Banach spaces. Annals of Math. 52 (1950), 518–527. · Zbl 0039.12202
[11] A. Kechris, A. Louveau: A classification of Baire class 1 functions. Trans. Amer. Math. Soc. 318 (1990), 209–236. · Zbl 0692.03031
[12] E. Kissin, V. Lomonosov, and V. Shulman: Implementation of derivations and invariant subspaces. Israel J. Math 134 (2003), 1–28. · Zbl 1047.47026
[13] G. Lancien: On the Szlenk index and the w*-dentability index. Quart. J. Math. Oxford 47 (1996), 59–71. · Zbl 0973.46014
[14] J. Lindenstrauss, C. Stegall: Examples of separable spaces which do not contain 1 and whose duals are nonseparable. Studia Math. 54 (1975), 81–105. · Zbl 0324.46017
[15] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces. I. Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92. Springer-Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0362.46013
[16] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces. II. Function Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 97. Springer-Verlag, Berlin-Heidelberg-New York, 1979. · Zbl 0403.46022
[17] E. Odell, H. P. Rosenthal: A double-dual characterization of separable Banach spaces containing 1. Israel J. Math. 20 (1975), 375–384. · Zbl 0312.46031
[18] A. Pelczynski: A connection between weakly unconditional convergence and weak completeness of Banach spaces. Bull. Acad. Pol. Sci. 6 (1958), 251–253. · Zbl 0082.10804
[19] C. Rickart: General Theory of Banach Algebras. Van Nostrand, 1960. · Zbl 0095.09702
[20] H. P. Rosenthal: A characterization of Banach spaces containing 1. Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. · Zbl 0297.46013
[21] H. P. Rosenthal: A characterization of Banach spaces containing c 0. J. Amer. Math. Soc. 7 (1994), 707–748. · Zbl 0824.46010
[22] A. Sersouri: On James’ type spaces. Trans. Amer. Math. Soc. 310 (1988), 715–745. · Zbl 0706.46021
[23] A. Sersouri: A note of the Lavrentiev index for quasi-reflexive Banach spaces. Banach space theory (Iowa City, IA, 1987). Contemp. Math. 85 (1989), 497–508.
[24] P. Wojtaszczyk: Banach spaces for analysts. Cambridge studies in Advanced Mathematics, Vol. 25. Cambridge University Press, 1991. · Zbl 0724.46012
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