Reciprocal Dunford--Pettis sets and V$^*$-sets in Banach lattices

Abstract

In this short note, we show that one cannot differentiate between reciprocal Dunford--Pettis sets and V$^*$-sets in a Banach lattice. That is, for a bounded subset $K$ of a Banach lattice $E$, $K$ is a V$^*$-set if and only if $K$ is a reciprocal Dunford--Pettis set, or equivalently, if and only if every disjoint sequence in the solid hull of $K$ is weakly null (i.e. $K$ is disjointly weakly compact).