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