On the Banach spaces with the property (V*) of Pelczynski

Summary We consider the Banach spaces with the property (V*) of Pelczynski giving a sufficient condition for a Banach space to have this property as well as a characterization of Banach lattices with the same property. Several other results are given which are concerning relationships among that property and other famous isomorphic properties of Banach spaces. Also a characterization of Banach spaces with property (V*) using Schauder decompositions is given. Some result concerning lifting of that property from a Banach space E to L¹(, E) is presented, too.Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.P.I. of Italy (40%).