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].
@article{10_4064_sm147_2_7,
author = {Miguel Mart{\'\i}n and Antonio M. Peralta},
title = {The alternative {Dunford{\textendash}Pettis} {Property} in
the predual of a von {Neumann} algebra},
journal = {Studia Mathematica},
pages = {197--200},
year = {2001},
volume = {147},
number = {2},
doi = {10.4064/sm147-2-7},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm147-2-7/}
}
TY - JOUR
AU - Miguel Martín
AU - Antonio M. Peralta
TI - The alternative Dunford–Pettis Property in
the predual of a von Neumann algebra
JO - Studia Mathematica
PY - 2001
SP - 197
EP - 200
VL - 147
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4064/sm147-2-7/
DO - 10.4064/sm147-2-7
LA - en
ID - 10_4064_sm147_2_7
ER -
%0 Journal Article
%A Miguel Martín
%A Antonio M. Peralta
%T The alternative Dunford–Pettis Property in
the predual of a von Neumann algebra
%J Studia Mathematica
%D 2001
%P 197-200
%V 147
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4064/sm147-2-7/
%R 10.4064/sm147-2-7
%G en
%F 10_4064_sm147_2_7
Miguel Martín; Antonio M. Peralta. The alternative Dunford–Pettis Property in
the predual of a von Neumann algebra. Studia Mathematica, Tome 147 (2001) no. 2, pp. 197-200. doi: 10.4064/sm147-2-7
