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A superreflexive Banach space which does not admit complex structure. (English) Zbl 0604.46019
An infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space is constructed; before only nonreflexive such examples were known. Also, a variant of Bourgain’s example of a complex Banach space with nonunique complex structure is presented. Other related results were since obtained by the author, see ”A Banach space without a basis, which has the bounded approximation property”, Acta. Math., in print.

MSC:
 46B20 Geometry and structure of normed linear spaces 46B10 Duality and reflexivity in normed linear and Banach spaces 47L05 Linear spaces of operators 52A22 Random convex sets and integral geometry (aspects of convex geometry)
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References:
 [1] S. F. Bellenot, Tsirelson superspaces and $${l_p}$$, Proc. Amer. Math. Soc. 94 (1985). [2] J. Bourgain, A complex Banach space such that $$X$$ and $$X$$ are not isomorphic, preprint. [3] T. Figiel, An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square, Studia Math. 42 (1972), 295 – 306. · Zbl 0213.12801 [4] Tadeusz Figiel, Stanisław Kwapień, and Aleksander Pełczyński, Sharp estimates for the constants of local unconditional structure of Minkowski spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 12, 1221 – 1226 (English, with Russian summary). · Zbl 0379.46012 [5] E. D. Gluskin, Finite-dimensional analogues of spaces without a basis, Dokl. Akad. Nauk SSSR 261 (1981), no. 5, 1046 – 1050 (Russian). · Zbl 0501.46013 [6] Robert C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518 – 527. · Zbl 0039.12202 [7] William B. Johnson, Banach spaces all of whose subspaces have the approximation property, Special topics of applied mathematics (Proc. Sem., Ges. Math. Datenverarb., Bonn, 1979) North-Holland, Amsterdam-New York, 1980, pp. 15 – 26. [8] N. J. Kalton and James W. Roberts, A rigid subspace of \?$$_{0}$$, Trans. Amer. Math. Soc. 266 (1981), no. 2, 645 – 654. · Zbl 0484.46004 [9] S. V. Kisljakov, Spaces with a ”small” annihilator, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 65 (1976), 192 – 195, 209 (Russian, with English summary). Investigations on linear operators and the theory of functions, VII. · Zbl 0347.46012 [10] D. R. Lewis, Finite dimensional subspaces of \?_{\?}, Studia Math. 63 (1978), no. 2, 207 – 212. · Zbl 0406.46023 [11] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. · Zbl 0259.46011 [12] P. Mankiewicz, Finite-dimensional Banach spaces with symmetry constant of order \sqrt \?, Studia Math. 79 (1984), no. 2, 193 – 200. · Zbl 0593.46017 [13] B. S. Mitjagin, The homotopy structure of a linear group of a Banach space, Uspehi Mat. Nauk 25 (1970), no. 5(155), 63 – 106 (Russian). I. Edelstein, B. Mitjagin, and E. Semenov, The linear groups of \? and \?$$_{1}$$ are contractible, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 27 – 33 (English, with Loose Russian summary). I. Edelstein, B. Mitjagin, and E. Semenov, Letter to the editors: ”The Linear groups of \? and \?$$_{1}$$ are contractible” (Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 27-33), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 213. B. S. Mitjagin and I. S. Èdel$$^{\prime}$$šteĭn, The homotopy type of linear groups of two classes of Banach spaces, Funkcional. Anal. i Priložen. 4 (1970), no. 3, 61 – 72 (Russian). [14] Gilles Pisier, Une nouvelle classe d’espaces de Banach vérifiant le théorème de Grothendieck, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, x, 69 – 90 (French, with English summary). · Zbl 0363.46019 [15] M. Rogalski, Sur le quotient volumique d’un espace de dimension finie, Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, pp. Comm. No. C3, 31 (French). · Zbl 0517.46014 [16] Stanisław Jerzy Szarek, On Kashin’s almost Euclidean orthogonal decomposition of \?\textonesuperior _{\?}, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 8, 691 – 694 (English, with Russian summary). · Zbl 0395.46015 [17] S. J. Szarek, Volume estimates and nearly Euclidean decompositions for normed spaces, Seminar on Functional Analysis, 1979 – 1980 (French), École Polytech., Palaiseau, 1980, pp. Exp. No. 25, 8. [18] Stanisław J. Szarek, The finite-dimensional basis problem with an appendix on nets of Grassmann manifolds, Acta Math. 151 (1983), no. 3-4, 153 – 179. · Zbl 0554.46004 [19] Stanisław J. Szarek, On the existence and uniqueness of complex structure and spaces with ”few” operators, Trans. Amer. Math. Soc. 293 (1986), no. 1, 339 – 353. · Zbl 0592.46016 [20] Stanisław Szarek and Nicole Tomczak-Jaegermann, On nearly Euclidean decomposition for some classes of Banach spaces, Compositio Math. 40 (1980), no. 3, 367 – 385. · Zbl 0432.46018
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