author={Mar'ia D. Acosta and Antonio M. Peralta},
  journal={Quarterly Journal of Mathematics},
We study a property weaker than the Dunford–Pettis property, introduced by Freedman, in the case of a JB*-triple. It is shown that a JBW*-triple W has this property if, and only if, 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 finitedimensional or Hilbert spaces (regarded as Cartan factor 1 or 4). 

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