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Slicely countably determined Banach spaces. (English) Zbl 1214.46004

This important paper introduces a new class of Banach spaces called slicely countably determined spaces.
The authors first define a slicely countably determined (SCD for short) set. A convex bounded subset \(A\) of a Banach space \(X\) is called SCD if there exists sequence of slices \(S_n\subset A\) such that each slice of \(A\) contains one of the \(S_n\); equivalently, \(A\) is SCD if there exists sequence of slices \(S_n\subset A\) such that, if \(B\subset A\) is a set intersecting all the \(S_n\), then \(A\) is contained in the closed convex hull of \(B\). Necessarily, an SCD-set is separable; examples of SCD sets include separable weakly compact sets or more generally RNP sets, and separable sets not containing \(\ell_1\)-bases. (The latter results relies on a theorem of Todorčevič on \(\pi\)-bases for the weak topology.) In the other direction, the unit ball of \(C[0,1]\) and \(L_1[0,1]\) (and, more generally, of any space with the Daugavet property) fails to be SCD.
Then, a separable Banach space \(X\) is called SCD if every bounded convex subset is SCD. Separable RNP-spaces and spaces not containing \(\ell_1\) are SCD, whereas spaces with the Daugavet property are not. The authors prove that \(X\) is SCD provided that a subspace \(Y\) and the quotient space \(X/Y\) are SCD, and they show that the unconditional sum \(\bigoplus_E X_n\) is SCD if each \(X_n\) is SCD and the sequence space \(E\) has a \(1\)-unconditional basis that is shrinking or boundedly complete; hence, \(c_0(\ell_1)\) and \(\ell_1(c_0)\) are examples of SCD spaces.
The authors give several applications to certain isometrical Banach space properties; for example, if \(B_X\) is SCD, then \(X\) has the alternative Daugavet property if and only if \(X\) is lush. This leads to the corollary that \(\ell_1\) embeds into \(X^*\) if the real Banach space \(X\) has numerical index \(1\).
One of the main results of the paper is a new approach to the results of [V. Kadets, R. Shvidkoy, G. Sirotkin and D. Werner, Trans.Am.Math.Soc.352, No. 2, 855–873 (2000; Zbl 0938.46016)] and [V. Kadets, R. Shvidkoy and D. Werner, Stud.Math.147, No. 3, 269–298 (2001; Zbl 0986.46010)]. For this, an operator version of SCD-ness is needed. An operator \(T:X\to Y\) is called SCD if \(T(B_X)\) is an SCD set, and it is called hereditarily SCD if every convex subset of \(T(B_X)\) is an SCD set. For example, strong Radon-Nikodým operators and operators not fixing a copy of \(\ell_1\) are hereditarily SCD if their ranges are separable. The authors prove that hereditarily SCD operators on Banach spaces with the Daugavet property are narrow, thus providing new and unified proofs of the main results of the above cited papers. By a separable reduction argument they also cover the case of nonseparable ranges.
Further, results on \(\pi\)-bases for the weak topology are discussed.
In a final section, a number of interesting problems are posed.

MSC:

46B04 Isometric theory of Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46B25 Classical Banach spaces in the general theory
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
47A12 Numerical range, numerical radius
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[1] Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. · Zbl 1094.46002
[2] S. Argyros, E. Odell, and H. Rosenthal, On certain convex subsets of \?\(_{0}\), Functional analysis (Austin, TX, 1986 – 87) Lecture Notes in Math., vol. 1332, Springer, Berlin, 1988, pp. 80 – 111. · Zbl 0658.46012 · doi:10.1007/BFb0081613
[3] Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. · Zbl 0946.46002
[4] J. BOURGAIN, La propriété de Radon-Nikodym, Publ. Math. de l’Univ. Pierre et Marie Curie 36, 1979. · Zbl 0423.46011
[5] J. Bourgain, Dentability and finite-dimensional decompositions, Studia Math. 67 (1980), no. 2, 135 – 148. · Zbl 0434.46017
[6] Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. · Zbl 0512.46017
[7] Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, and Javier Merí, Properties of lush spaces and applications to Banach spaces with numerical index 1, Studia Math. 190 (2009), no. 2, 117 – 133. · Zbl 1168.46005 · doi:10.4064/sm190-2-2
[8] Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, and Dirk Werner, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 93 – 102. · Zbl 1121.47001 · doi:10.1017/S0305004106009650
[9] Jesús M. F. Castillo and Manuel González, Three-space problems in Banach space theory, Lecture Notes in Mathematics, vol. 1667, Springer-Verlag, Berlin, 1997. · Zbl 0914.46015
[10] Gustave Choquet, Lectures on analysis. Vol. II: Representation theory, Edited by J. Marsden, T. Lance and S. Gelbart, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0181.39602
[11] Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. · Zbl 0782.46019
[12] D. van Dulst, Characterizations of Banach spaces not containing \?\textonesuperior , CWI Tract, vol. 59, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. · Zbl 0684.46017
[13] J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, The numerical index of a normed space, J. London Math. Soc. (2) 2 (1970), 481 – 488. · Zbl 0197.10402
[14] N. Ghoussoub, G. Godefroy, B. Maurey, and W. Schachermayer, Some topological and geometrical structures in Banach spaces, Mem. Amer. Math. Soc. 70 (1987), no. 378, iv+116. · Zbl 0651.46017 · doi:10.1090/memo/0378
[15] Yevgen Ivakhno, Vladimir Kadets, and Dirk Werner, The Daugavet property for spaces of Lipschitz functions, Math. Scand. 101 (2007), no. 2, 261 – 279. · Zbl 1177.46010 · doi:10.7146/math.scand.a-15044
[16] V. KADETS, M. MARTíN, J. MERí, AND R. PAYá, Convexity and smoothness of Banach spaces with numerical index one, Illinois J. Math. 53 (2009), 163-182. · Zbl 1197.46008
[17] Vladimir Kadets, Miguel Martín, Javier Merí, and Varvara Shepelska, Lushness, numerical index one and duality, J. Math. Anal. Appl. 357 (2009), no. 1, 15 – 24. · Zbl 1183.46009 · doi:10.1016/j.jmaa.2009.03.055
[18] Vladimir Kadets, Miguel Martín, and Rafael Payá, Recent progress and open questions on the numerical index of Banach spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 155 – 182 (English, with English and Spanish summaries). · Zbl 1111.46007
[19] Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855 – 873. · Zbl 0938.46016
[20] Vladimir M. Kadets, Roman V. Shvidkoy, and Dirk Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), no. 3, 269 – 298. · Zbl 0986.46010 · doi:10.4064/sm147-3-5
[21] Vladimir Kadets and Dirk Werner, A Banach space with the Schur and the Daugavet property, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1765 – 1773. · Zbl 1043.46015
[22] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. · Zbl 0362.46013
[23] Ginés López, Miguel Martín, and Rafael Payá, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), no. 2, 207 – 212. · Zbl 0921.46015 · doi:10.1112/S002460939800513X
[24] Miguel Martín, The alternative Daugavet property of \?*-algebras and \?\?*-triples, Math. Nachr. 281 (2008), no. 3, 376 – 385. · Zbl 1146.46045 · doi:10.1002/mana.200510608
[25] Miguel Martín and Timur Oikhberg, An alternative Daugavet property, J. Math. Anal. Appl. 294 (2004), no. 1, 158 – 180. · Zbl 1054.46010 · doi:10.1016/j.jmaa.2004.02.006
[26] Haskell Rosenthal, On the structure of nondentable closed bounded convex sets, Adv. in Math. 70 (1988), no. 1, 1 – 58. · Zbl 0654.46024 · doi:10.1016/0001-8708(88)90050-3
[27] Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. · Zbl 1090.46001
[28] W. Schachermayer, An example concerning strong regularity and points of continuity in Banach spaces, Functional analysis (Austin, TX, 1986 – 87) Lecture Notes in Math., vol. 1332, Springer, Berlin, 1988, pp. 64 – 79. · Zbl 0649.46011 · doi:10.1007/BFb0081612
[29] R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), no. 2, 198 – 212. · Zbl 0964.46006 · doi:10.1006/jfan.2000.3626
[30] Stevo Todorčević, Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 (1999), no. 4, 1179 – 1212. · Zbl 0938.26004
[31] Lutz Weis, On the surjective (injective) envelope of strictly (co-) singular operators, Studia Math. 54 (1975/76), no. 3, 285 – 290. · Zbl 0323.47016
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