Properties (u) and (V*) of Pelczynski in symmetric spaces of T-measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V^*) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.