The alternative Dunford–Pettis Property in the predual of a von Neumann algebra
Tom 147 / 2001
Studia Mathematica 147 (2001), 197-200
MSC: 46L05, 46L10.
DOI: 10.4064/sm147-2-7
Streszczenie
Let $A$ be a type II von Neumann algebra with predual $A_{*}$. We prove that $A_{*}$ does not have the alternative Dunford–Pettis property introduced by W. Freedman [7], i.e., there is a sequence $(\varphi _{n})$ converging weakly to $\varphi $ in $A_{*}$ with $\| \varphi _{n}\| =\| \varphi \| =1$ for all $n\in {\mathbb N}$ and a weakly null sequence $(x_{n})$ in $A$ such that $\varphi _{n} (x_{n}) \nrightarrow 0$. This answers a question posed in [7].
