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

Abstract

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.