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A continuous image of a Radon-Nikodým compact space which is not Radon-Nikodým. (English) Zbl 1285.46013

Summary: We construct a continuous image of a Radon-Nikodým compact space which is not Radon-Nikodým compact, solving the problem posed in the 1980s by I. Namioka [Mathematika 34, No. 2, 258–281 (1987; Zbl 0654.46017)].

MSC:

46B26 Nonseparable Banach spaces
54D30 Compactness

Citations:

Zbl 0654.46017
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References:

[1] A. V. Arkhangel’skiĭ, Topological Function Spaces , Math. Appl. (Soviet Ser.) 78 , Kluwer Academic, Dordrecht, 1992.
[2] A. V. Arkhangel’skiĭ, General Topology, II , Encyclopaedia Math. Sci. 50 , Springer, Berlin, 1996.
[3] A. V. Arhangel’skiĭ, Topological vector spaces, compacta, and unions of subspaces , J. Math. Anal. Appl. 350 (2009), 616-622. · Zbl 1163.54018
[4] A. D. Arvanitakis, Some remarks on Radon-Nikodým compact spaces , Fund. Math. 172 (2002), 41-60. · Zbl 1012.46021
[5] A. D. Arvanitakis and A. Avilés, Some examples of continuous images of Radon-Nikodým compact spaces , Czechoslovak Math. J. 59(134) (2009), 1027-1038. · Zbl 1224.46030
[6] A. Avilés, Radon-Nikodým compact spaces of low weight and Banach spaces , Studia Math 166 (2005), 71-82. · Zbl 1086.46014
[7] A. Avilés, Linearly ordered Radon-Nikodým compact spaces , Topology Appl. 154 (2007), 404-409. · Zbl 1109.54022
[8] A. Avilés and O. F. K. Kalenda, Compactness in Banach space theory-selected problems , Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104 (2010), 337-352. · Zbl 1227.46016
[9] Y. Benyamini, M. E. Rudin, M. Wage, Continuous images of weakly compact subsets of Banach spaces , Pacific J. Math. 70 (1977), 309-324. · Zbl 0374.46011
[10] K. Ciesielski and R. Pol, A weakly Lindelöf function space C(K) without any continuous injection into \(c_{0}(\Gamma)\) , Bull. Pol. Acad. Sci. Math. 32 (1984), 681-688. · Zbl 0571.54014
[11] W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators . J. Funct. Anal. 17 (1974), 311-327. · Zbl 0306.46020
[12] J. Diestel and J. J. Uhl, Vector Measures , Math. Surveys Monogr. 15 , American Mathematical Society, Providence, 1977.
[13] M. J. Fabian, Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund spaces , Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Interscience, Wiley, New York, 1997. · Zbl 0883.46011
[14] M. J. Fabian, “Overclasses of the class of Radon-Nikodým compact spaces” in Methods in Banach Space Theory , London Math. Soc. Lecture Note Ser. 337 , Cambridge Univ. Press, Cambridge, 2006, 197-214. · Zbl 1149.46017
[15] M. J. Fabian, M. Heisler, and E. Matoušková, Remarks on continuous images of Radon-Nikodým compacta , Comment. Math. Univ. Carolin. 39 (1998), 59-69. · Zbl 0937.46015
[16] M. J. Fabian, V. Montesinos, and V. Zizler, Weak compactness and \(\sigma\)-Asplund generated Banach spaces , Studia Math. 181 (2007), 125-152. · Zbl 1127.46003
[17] V. V. Fedorchuk, Bicompacta with noncoinciding dimensionalities (in Russian), Dokl. Akad. Nauk SSSR 182 (1968), 275-277; English translation in Soviet Math. Dokl. 9 (1968), 1148-1150. · Zbl 0186.27003
[18] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires , Mem. Amer. Math. Soc. 1955 (1955), no. 16. · Zbl 0123.30301
[19] S. P. Gul’ko, Properties of sets that lie in \(\Sigma\)-products , Sov. Math. Dokl. 18 (1977), 1438-1442.
[20] M. Iancu and S. Watson, On continuous images of Radon-Nikodým compact spaces through the metric characterization , Topology Proc. 26 (2001/02), 677-693. · Zbl 1083.54006
[21] J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps , Acta Math. 155 (1985), 41-79. · Zbl 0588.54020
[22] P. Koszmider, Banach spaces of continuous functions with few operators , Math. Ann. 330 (2004), 151-183. · Zbl 1064.46009
[23] P. Koszmider, A survey on Banach spaces \(C(K)\) with few operators , Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104 (2010), 309-326. · Zbl 1227.46021
[24] E. Matoušková and C. Stegall, “Compact spaces with a finer metric topology and Banach spaces” in General Topology in Banach Spaces , Nova Sci., Huntington, New York, 2001, 81-101. · Zbl 1035.46014
[25] E. Michael and M. E. Rudin, Another note on Eberlein compacts , Pacific J. Math. 72 (1977), 497-499. · Zbl 0344.54018
[26] I. Namioka, Radon-Nikodým compact spaces and fragmentability , Mathematika 34 (1987), 258-281. · Zbl 0654.46017
[27] I. Namioka, On generalizations of Radon-Nikodým compact spaces , Topology Proc. 26 (2001/02), 741-750. · Zbl 1083.54012
[28] I. Namioka, Fragmentability in Banach spaces: Interaction of topologies , Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Math. RACSAM 104 (2010), 283-308. · Zbl 1237.54002
[29] I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces , Duke Math. J. 42 (1975), 735-750. · Zbl 0332.46013
[30] J. Orihuela, W. Schachermayer, and M. Valdivia, Every Radon-Nikodým Corson compact space is Eberlein compact , Studia Math. 98 (1991), 157-174. · Zbl 0771.46015
[31] G. Plebanek, A construction of a Banach space \(C(K)\) with few operators , Topology Appl. 143 (2004), 217-239. · Zbl 1064.46011
[32] R. Pol, A function space \(C(X)\) which is weakly Lindelöf but not weakly compactly generated , Studia Math. 64 (1979), 279-285. · Zbl 0424.46011
[33] O. I. Reynov, On a class of Hausdorff compacts and GSG Banach spaces , Studia Math. 71 (1981/82), 113-126. · Zbl 0415.46013
[34] H. P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces , Compos. Math. 28 (1974), 83-111. · Zbl 0298.46013
[35] I. Schlackow, Centripetal operators and Koszmider spaces , Topology Appl. 155 (2008), 1227-1236. · Zbl 1151.46017
[36] C. Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodým property , Israel J. Math. 29 (1978), 408-412. · Zbl 0374.46015
[37] C. Stegall, The Radon-Nikodým property in conjugate Banach spaces, II , Trans. Amer. Math. Soc. 264 (1981), no. 2, 507-519. · Zbl 0475.46016
[38] C. Stegall, “Spaces of Lipschitz functions on Banach spaces” in Functional Analysis (Essen, 1991) , Lect. Notes Pure Appl. Math. 150 , Dekker, New York, 1994, 265-278. · Zbl 0821.46021
[39] S. Watson, “The construction of topological spaces: Planks and resolutions” in Recent Progress in General Topology (Prague, 1991) , North-Holland, Amsterdam, 1992, 673-757. · Zbl 0803.54001
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