Dunford-Pettis type properties in \(L_1\) of a vector measure (Q6577504)

The Banach lattices \(L_1(\nu)\) of a countably additive vector measure \(\nu\) with values in a Banach space \(X\) has attracted lots of attention over the last decades. Their importance is witnessed by the fact, due to Curbera, that every Banach lattice with order continuous norm and a weak unit can be viewed as such a space.\N\NAs a rule of thumb, properties of \(L_1(\nu)\) are reflected in (direct or indirect) properties of the measure \(\nu\). In this paper the author gives a thorough overview over what is known concerning the interrelated problems:\N\begin{itemize}\N\item When is \(L_1(\nu)\) lattice isomorphic to an AL-space?\N\item When does \(L_1(\nu)\) have the positive Schur property?\N\item When does \(L_1(\nu)\) have,the Dunford-Pettis property?\N\end{itemize}\NThe governing open question is the following: Suppose the integration operator \(I_\nu : L_1(\nu)\to X\) given by \(I_\nu(f)=\int f\,d\nu\) is Dunford-Pettis and \(X\) does not contain \(\ell_1\). Is then \(L_1(\nu)\) lattice isomorphic to an AL-space? The answer is known to be yes with the stronger condition that \(I_\nu\) factors through an Asplund space, see the author's paper [\textit{J.~Rodríguez}, Colloq. Math. 144, No.~1, 115--125 (2016; Zbl 1355.46033)].\N\NPushing forward towards an answer to the main question involves rather deep results and techniques. Section~3 of the paper presents the machinery leading to its partial solution. In Section~4 a new characterization of the positive Schur property of \(L_1(\nu)\) in terms of a certain Dunford-Pettis property is given, and a new proof of the following result of \N\textit{G.~P. Curbera} [Proc. Am. Math. Soc. 123, No.~12, 3797--3806 (1995; Zbl 0848.46015)]\N is given: \(L_1(\nu)\) has the Dunford-Pettis property if it has the positive Schur property and \(\nu\) is of \(\sigma\)-finite variation. It is then stated as an open problem whether the assumption on the variation can be dropped.\N\NThe paper ends with presenting a class of \(\nu\)'s such that \(L_1(\nu)\) enjoys both the positive Schur property and the Dunford-Pettis property, but is not lattice-isomorphic to any AL-space.