An alternative Dunford-Pettis property for JB*-triples

We study a property weaker than the Dunford-Pettis property, introduced by W. Freedman, in the case of a JB*-triple. It is shown that a JBW*-triple W has this property if, and only if, either W is a Hilbert space (regarded as a type 1 or 4 Cartan factor) or W has the Dunford-Pettis property. As a consequence, we get that the JBW*-triples satisfying the Kadec-Klee property are either finite-dimensional or Hilbert spaces (regarded as Cartan factor 1 or 4).
María D. Acosta <dacosta@goliat.ugr.es> Antonio M. Peralta <aperalta@goliat.ugr.es>