Previous |  Up |  Next

Article

Keywords:
Gelfand-Phillips property; Mazur property; generalized density
Summary:
We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space $E$ can be naturally expressed in terms of {\it weak}* continuity of seminorms on the unit ball of $E^*$. \endgraf We attempt to carry out a construction of a Banach space of the form $C(K)$ which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the {\it weak}* sequential closure of atomic measures, and the set-theoretic properties of generalized densities on the natural numbers.
References:
[1] Balcar, B., Pelant, J., Simon, P.: The space of ultrafilters on N covered by nowhere dense sets. Fundam. Math. 110 (1980), 11-24. MR 0600576 | Zbl 0568.54004
[2] Rao, K. P. S. Bhaskara, Rao, M. Bhaskara: Theory of Charges. Academic Press London (1983). MR 0751777
[3] Blass, A.: Combinatorial cardinal characteristics of the continuum. (to appear) as a chapter in Handbook of Set Theory. MR 2768685
[4] Borodulin-Nadzieja, P.: On measures on minimally generated Boolean algebras. Topology Appl. 154 (2007), 3107-3124. DOI 10.1016/j.topol.2007.03.014 | MR 2364639
[5] Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 119 (1984), 55-58. DOI 10.1002/mana.19841190105 | MR 0774176 | Zbl 0601.47019
[6] Drewnowski, L.: On Banach spaces with the Gelfand-Phillips property. Math. Z. 193 (1986), 405-411. DOI 10.1007/BF01229808 | MR 0862887 | Zbl 0629.46020
[7] Edgar, G. A.: Measurability in a Banach space II. Indiana Univ. Math. J. 28 (1979), 559-579. DOI 10.1512/iumj.1979.28.28039 | MR 0542944 | Zbl 0418.46034
[8] Farkas, B., Soukup, L.: More on cardinal invariants of analytic $P$-ideals. Preprint. MR 2537837
[9] Freedman, W.: An extension property for Banach spaces. Colloq. Math. 91 (2002), 167-182. DOI 10.4064/cm91-2-2 | MR 1898630 | Zbl 1028.46020
[10] Howard, J.: On {\it weak}* separable subsets of dual Banach spaces. Missouri J. Math. Sci 7 (1995), 116-118. MR 1455281
[11] Hernandez-Hernandez, F., Hrusák, M.: Cardinal invariants of $P$-ideals. Preprint.
[12] Kalenda, O.: Valdivia compact spaces in topology and Banach space theory. Extr. Math. 15 (2000), 1-85. MR 1792980 | Zbl 0983.46021
[13] Kalenda, O.: (I)-envelopes of unit balls and James' characterization of reflexivity. Stud. Math. 182 (2007), 29-40. DOI 10.4064/sm182-1-2 | MR 2326490 | Zbl 1139.46018
[14] Koppelberg, S.: Minimally generated Boolean algebras. Order 5 (1989), 393-406. DOI 10.1007/BF00353658 | MR 1010388 | Zbl 0676.06019
[15] Koppelberg, S.: Counterexamples in minimally generated Boolean algebras. Acta Univ. Carol. Math. Phys. 29 (1988), 27-36. MR 0983448 | Zbl 0676.06020
[16] Koszmider, P.: Forcing minimal extensions of Boolean algebras. Trans. Am. Math. Soc. 351 (1999), 3073-3117. DOI 10.1090/S0002-9947-99-02145-5 | MR 1467471 | Zbl 0922.03071
[17] Leung, D. H.: A Gelfand-Phillips property with respect to the weak topology. Math. Nachr. 149 (1990), 177-181. DOI 10.1002/mana.19901490114 | MR 1124803 | Zbl 0765.46007
[18] Leung, D. H.: On Banach spaces with Mazur's property. Glasg. Math. J. 33 (1991), 51-54. DOI 10.1017/S0017089500008028 | MR 1089953 | Zbl 0745.46021
[19] Mazur, S.: On continuous mappings on Cartesian products. Fundam. Math. 39 (1952), 229-238. MR 0055663
[20] Mercourakis, S.: Some remarks on countably determined measure and uniform distribution of sequences. Monatsh. Math. 121 (1996), 79-111. DOI 10.1007/BF01299640 | MR 1375642
[21] Plebanek, G.: On the space of continuous functions on a dydadic set. Mathematika 38 (1991), 42-49. DOI 10.1112/S0025579300006422 | MR 1116683
[22] Plebanek, G.: On some properties of Banach spaces of continuous functions. Séminaire d'initiation a l'analyse 1991/92, Vol. 31 G. Choquet et al. Université Pierre et Marie Curie Paris (1994). Zbl 0876.46016
[23] Plebanek, G.: On Mazur property and realcompactness in $C(K)$. In: Topology, Measure and Fractals, Math. Res. Vol. 66 C. Bandt et al. Akademie Verlag (1992). MR 1226275 | Zbl 0850.46019
[24] Plebanek, G.: On Pettis integrals with separable range. Colloq. Math. 64 (1993), 71-78. MR 1201444 | Zbl 0823.28005
[25] Plebanek, G.: Compact spaces that result from adequate families of sets. Topology Appl. 65 (1995), 257-270 Erratum: Topology Appl. 72 (1996), 99. DOI 10.1016/0166-8641(95)00006-3 | MR 1357868 | Zbl 0869.54003
[26] Sinha, D. P., Arora, K. K.: On the Gelfand-Phillips property in Banach spaces with PRI. Collect. Math. 48 (1997), 347-354. MR 1475810 | Zbl 0903.46015
[27] Talagrand, M.: Pettis integral and measure theory. Mem. Am. Math. Soc. 307 (1984). MR 0756174 | Zbl 0582.46049
[28] Schlumprecht, T.: Limited sets in $C(K)$-spaces and examples concerning the Gelfand-Phillips property. Math. Nachr. 157 (1992), 51-64. DOI 10.1002/mana.19921570105 | MR 1233046 | Zbl 0797.46013
[29] Wilansky, A.: Mazur spaces. Int. J. Math. Sci. 4 (1981), 39-53. DOI 10.1155/S0161171281000021 | MR 0606656 | Zbl 0466.46005
Partner of
EuDML logo