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algebras of operators with only one non-trivial invariant subspace; invariant subspaces under the action of the algebra of biconjugates operators; transitivity; property (u) of Pelczynski
We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $\mathbb{A}(X)$ of biconjugates of bounded operators on $X$: $\mathbb{A}(X)=\lbrace T^{**}\: T \in \mathcal {B}(X)\rbrace $. 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 property (u) of Pelczynski is simple.
[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. DOI 10.1016/S1874-5849(03)80030-X | MR 1999190
[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. DOI 10.4153/CJM-1975-131-3 | MR 0399817
[3] G. Godefroy, D. Li: Banach spaces which are $M$-ideals in their bidual have property  (u). Ann. Inst. Fourier. 39 (1989), 361–371. DOI 10.5802/aif.1170 | MR 1017283 | Zbl 0659.46014
[4] G. Godefroy, P. Saab: Weakly unconditionnaly converging series in $M$-ideals. Math. Scand. 64 (1989), 307–318. MR 1037465
[5] G. Godefroy, M. Talagrand: Nouvelles classes d’espaces de Banach à prédual unique. Séminaire d’analyse fonctionnelle École Polytechnique, Exposé  $n^\circ 9$ (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. MR 0983385
[7] G. Godefroy, N. J. Kalton, and P. D. Saphar: Unconditional ideals in Banach spaces. Studia Math. 104 (1993), 13–59. MR 1208038
[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. MR 1831176
[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. MR 1126734
[10] R. C. James: Bases and reflexivity of Banach spaces. Annals of Math. 52 (1950), 518–527. DOI 10.2307/1969430 | MR 0039915 | Zbl 0039.12202
[11] A. Kechris, A. Louveau: A classification of Baire class 1  functions. Trans. Amer. Math. Soc. 318 (1990), 209–236. MR 0946424
[12] E. Kissin, V. Lomonosov, and V. Shulman: Implementation of derivations and invariant subspaces. Israel J.  Math 134 (2003), 1–28. DOI 10.1007/BF02787401 | MR 1972173
[13] G. Lancien: On the Szlenk index and the $w^{*}$-dentability index. Quart. J.  Math. Oxford 47 (1996), 59–71. DOI 10.1093/qmath/47.1.59 | MR 1380950
[14] J. Lindenstrauss, C. Stegall: Examples of separable spaces which do not contain  $\ell _1$ and whose duals are nonseparable. Studia Math. 54 (1975), 81–105. MR 0390720
[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. MR 0500056
[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. MR 0540367
[17] E. Odell, H. P. Rosenthal: A double-dual characterization of separable Banach spaces containing  $\ell _{1}$. Israel J.  Math. 20 (1975), 375–384. DOI 10.1007/BF02760341 | MR 0377482
[18] A. Pelczynski: A connection between weakly unconditional convergence and weak completeness of Banach spaces. Bull. Acad. Pol. Sci. 6 (1958), 251–253. MR 0115072
[19] C. Rickart: General Theory of Banach Algebras. Van Nostrand, , 1960. MR 0115101 | Zbl 0095.09702
[20] H. P. Rosenthal: A characterization of Banach spaces containing  $\ell _{1}$. Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. DOI 10.1073/pnas.71.6.2411 | MR 0358307
[21] H. P. Rosenthal: A characterization of Banach spaces containing  $c_{0}$. J.  Amer. Math. Soc. 7 (1994), 707–748. MR 1242455
[22] A. Sersouri: On James’ type spaces. Trans. Amer. Math. Soc. 310 (1988), 715–745. MR 0973175 | 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. MR 0983401
[24] P. Wojtaszczyk: Banach spaces for analysts. Cambridge studies in Advanced Mathematics, Vol. 25. Cambridge University Press, 1991. MR 1144277
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