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Keywords:
Pełczyński's property (V); $C^*$-algebra; Grothendieck property
Summary:
A Banach space $X$ has Pełczyński's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. H. Pfitzner proved that $C^*$-algebras have Pełczyński's property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner's theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.
References:
[1] Angosto, C., Cascales, B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156 (2009), 1412-1421. DOI 10.1016/j.topol.2008.12.011 | MR 2502017 | Zbl 1176.46012
[2] Behrends, E.: New proofs of Rosenthal's $\ell^1$-theorem and the Josefson-Nissenzweig theorem. Bull. Pol. Acad. Sci., Math. 43 (1995), 283-295. MR 1414785 | Zbl 0847.46007
[3] Bendová, H.: Quantitative Grothendieck property. J. Math. Anal. Appl. 412 (2014), 1097-1104. DOI 10.1016/j.jmaa.2013.11.033 | MR 3147271 | Zbl 1322.46008
[4] Blasi, F. S. De: On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 21 (1977), 259-262. MR 0482402 | Zbl 0365.46015
[5] Gasparis, I.: $\epsilon$-weak Cauchy sequences and a quantitative version of Rosenthal's $\ell_1$-theorem. J. Math. Anal. Appl. 434 (2016), 1160-1165. DOI 10.1016/j.jmaa.2015.09.079 | MR 3415714 | Zbl 06509536
[6] Harmand, P., Werner, D., Werner, W.: $M$-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics 1547, Springer, Berlin (1993). DOI 10.1007/BFb0084355 | MR 1238713 | Zbl 0789.46011
[7] Kalenda, O. F. K., Pfitzner, H., Spurný, J.: On quantification of weak sequential completeness. J. Funct. Anal. 260 (2011), 2986-2996. DOI 10.1016/j.jfa.2011.02.006 | MR 2774062 | Zbl 1248.46012
[8] Krulišová, H.: Quantification of Pe{ł}czyński's property (V). To appear in Math. Nachr.
[9] Lechner, J.: 1-Grothendieck $C(K)$ spaces. J. Math. Anal. Appl. 446 (2017), 1362-1371. DOI 10.1016/j.jmaa.2016.06.038 | MR 3563039 | Zbl 1364.46015
[10] Pfitzner, H.: Weak compactness in the dual of a $C^{\ast}$-algebra is determined commutatively. Math. Ann. 298 (1994), 349-371. DOI 10.1007/BF01459739 | MR 1256621 | Zbl 0791.46035
[11] Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987). MR 0924157 | Zbl 0925.00005
[12] Simons, S.: On the Dunford-Pettis property and Banach spaces that contain $c_0$. Math. Ann. 216 (1975), 225-231. DOI 10.1007/BF01430962 | MR 0402470 | Zbl 0294.46010
[13] Takesaki, M.: Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences 124, Operator Algebras and Non-Commutative Geometry 5, Springer, Berlin (2002). MR 1873025 | Zbl 0990.46034
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