THE ALTERNATIVE DUNFORD–PETTIS PROPERTY, CONJUGATIONS AND REAL FORMS OF C∗-ALGEBRAS

Let τ be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space X and let Xτ be the real form of X of τ-fixed points. In contrast to the Dunford–Pettis property, the alternative Dunford–Pettis property need not lift from Xτ to X. If X is a C*-algebra it is shown that Xτ has the alternative Dunford–Pettis property if and only if X does and an analogous result is shown when X is the dual space of a C*-algebra. One consequence is that both Dunford–Pettis properties coincide on all real forms of C*-algebras.