Czechoslovak Mathematical Journal, online first, 15 pp.

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová

Received May 16, 2016.   First published August 14, 2017.

Hana Krulišová, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: krulisova@karlin.mff.cuni.cz

Abstract: 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.

Keywords: Pełczyński's property (V); $C^*$-algebra; Grothendieck property

Classification (MSC 2010): 46B04, 46L05, 47B10

DOI: 10.21136/CMJ.2017.0242-16

Full text available as PDF.


References:
[1] C. Angosto, B. Cascales: 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] E. Behrends: 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] H. Bendová: 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] F. S. De Blasi: 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] I. Gasparis: $\epsilon$-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] P. Harmand, D. Werner, W. Werner: $M$-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] O. F. K. Kalenda, H. Pfitzner, J. Spurný: 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] H. Krulišová: Quantification of Pełczyński's property (V). To appear in Math. Nachr.
[9] J. Lechner: 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] H. Pfitzner: 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] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York (1987). MR 0924157 | Zbl 0925.00005
[12] S. Simons: 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] M. Takesaki: 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


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.


[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]