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

Volume 147 / 2001

Miguel Martín, Antonio M. Peralta Studia Mathematica 147 (2001), 197-200 MSC: 46L05, 46L10. DOI: 10.4064/sm147-2-7

Abstract

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].

Authors

  • Miguel MartínDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail
  • Antonio M. PeraltaDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail

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