Geodesic
C*-algebras have a quantitative version of Pełczyński's property (V)
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 937-951
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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.
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.
DOI : 10.21136/CMJ.2017.0242-16
Classification : 46B04, 46L05, 47B10
Keywords: Pełczyński's property (V); $C^*$-algebra; Grothendieck property 
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Krulišová, Hana. C*-algebras have a quantitative version of Pełczyński's property (V). Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 937-951. doi: 10.21136/CMJ.2017.0242-16

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