The alternative Dunford--Pettis property in the predual of a von Neumann algebra (Q2759174)

A Banach space \(E\) has the alternative Dunford-Pettis property if, for any sequence \((a_n)\) in the dual \(E^*\) of \(E\) converging weakly to zero, and any sequence \((x_n)\) of elements of \(E\) of norm one converging weakly to an element \(x\) of norm one, the sequence (\((a_n(x_n))\) converges to zero.NEWLINENEWLINENEWLINEIn this paper the authors use the fact that every W\(^*\)-algebra of type \(\text{II}_1\) contains a countably infinite-dimensional spin factor to show that the predual of a W\(^*\)-algebra of type II does not have the alternative Dunford-Pettis property.