Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 937-951, 2017


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

Hana Krulišová

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.
Keywords:  Pełczyński's property (V); $C^*$-algebra; Grothendieck property
Classification MSC:  46B04, 46L05, 47B10
DOI:  10.21136/CMJ.2017.0242-16

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Affiliations:   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

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