Complemented copies of $\ell_1$ in spaces of vector valued measures and applications

Abstract

Let $X$ be a Banach space and $(Ω,Σ)$ be a measure space. We provide a characterization of sequences in the space of $X$-valued countably additive measures on $Ω,Σ)$ of bounded variation that generate complemented copies of $\ell_1$. As application, we prove that if a dual Banach space $E^*$ has Pełczyński's property (V*) then so does the space of $E^*$-valued countably additive measures with bounded variation. Another application, we show that for a Banach space $X$, the space $\ell_\infty(X)$ contains a complemented copy of $\ell_1$ if and only if $X$ contains all $\ell_1^n$ uniformly complemented.