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:

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.

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