The alternative Dunford–Pettis Property in the predual of a von Neumann algebra

Miguel Martín ¹   ; Antonio M. Peralta ¹

¹ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 147 (2001) no. 2, pp. 197-200

Résumé

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].

Détail

DOI : 10.4064/sm147-2-7

Keywords: type von neumann algebra predual * prove * does have alternative dunford pettis property introduced freedman there sequence varphi converging weakly varphi * varphi varphi mathbb weakly null sequence varphi nrightarrow answers question posed