Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces

Abstract

Let $X$ be a Banach space and $(f_n)_n$ be a bounded sequence in $L^1(X)$. We prove a complemented version of the celebrated Talagrand's dichotomy i.e we show that if $(e_n)_n$ denotes the unit vector basis of $c_0$, there exists a sequence $g_n \in \text{conv}(f_n,f_{n+1},\dots)$ such that for almost every $\omega$, either the sequence $(g_n(\omega) \otimes e_n)$ is weakly Cauchy in $X \widehat{\otimes}_\pi c_0$ or it is equivalent to the unit vector basis of $\ell^1$. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of $\ell^1$ in $L^1(X)$. As an application, we show that for a Banach space $X$, the space $L^1(X)$ has Pe\l czy\'nski's property $(V^*)$ if and only if $X$ does.