The Dunford-Pettis property for symmetric spaces

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that l(1), c(o) and l(infinity) are the only symmetric sequence spaces with the Dunford-Pettis property, and that in the class of symmetric spaces on (0, alpha), 0 < alpha less than or equal to infinity, the only spaces with the Dunford-Pettis property are L-1, L-infinity, L-1 boolean AND L-infinity, L-1 + L-infinity, (L-infinity)(o) and (L-1 + L-infinity)(o),where X-o denotes the norm closure of L-1 boolean AND L-infinity in X. It is also proved that all Banach dual spaces of L-1 boolean AND L-infinity and L-1 + L-infinity have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces (L-1 + L-infinity)(o) and (L-infinity)(o) have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Kothe-Bochner spaces.