The alternative Dunford-Pettis property, conjugations and real forms of C*-algebras

Let tau be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space X and let X-tau be the real form of,X of tau-fixed points. In contrast to the Dunford-Pettis property, the alternative Dunford-Pettis property need not lift from X-tau to X. If X is a C*-algebra it is shown that X-tau 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.