Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 937-951, 2017
C*-algebras have a quantitative version of Pełczyński's property (V)
Received May 16, 2016. First published August 14, 2017.
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.
Affiliations: Hana Krulišová, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: firstname.lastname@example.org