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

TL;DR

This paper extends the understanding of Pelczynski's property (V) by providing a quantitative version for C*-algebras and related spaces, linking it to the Grothendieck property in dual spaces.

Contribution

It generalizes Pfitzner's theorem to a quantitative setting and connects the property (V) with the Grothendieck property in dual Banach spaces.

Findings

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

Quantitative property (V) implies a quantitative Grothendieck property in dual spaces

Extension of Pfitzner's theorem to a broader class of spaces

Abstract

A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T: X -> Y is weakly compact. H. Pfitzner proved that C*-algebras have Pelczynski's property (V). In the preprint "H. Krulisova: Quantification of Pelczynski's property (V)" 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.