The alternative Dunford-Pettis property in $C^*$-algebras and von Neumann preduals

Abstract:

A Banach space $X$ is said to have the alternative Dunford-Pettis property if, whenever a sequence $x_{n} \rightarrow x$ weakly in $X$ with $\|x_{n}\| \rightarrow \|x\|$, we have $\rho _{n} (x_{n}) \rightarrow 0$ for each weakly null sequence $(\rho _{n})$ in X$^*$. We show that a $C^*$-algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for $C^*$-algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.