On weak* convergent sequences in duals of symmetric spaces of \tau-measurable operators

Abstract:

 We answer in affirmative the open question raised by Figiel, Johnson and Pelczynski on whether the predual of a $\sigma$-finite von Neumann algebra has property $(k)$. Our approach here is to show that the Mackey topology on a $\sigma$-finite von Neumann algebra with respect to its predual is metrizable on norm bounded sets.  This,  in turn, rests on combining the classical criterion of Akemann that each weakly relatively compact subset of the predual of a von Neumann algebra is of uniformly absolutely continuous norm (equi-integrable) with a characterisation of such sets in the predual of a  $\sigma$-finite von Neumann algebra, due to Raynaud and Xu.