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


Authors: Leslie J. Bunce and Antonio M. Peralta
Journal: Proc. Amer. Math. Soc. 131 (2003), 1251-1255
MSC (2000): Primary 46B04, 46B20, 46L05, 46L10
DOI: https://doi.org/10.1090/S0002-9939-02-06700-X
Published electronically: September 5, 2002
MathSciNet review: 1948117
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Abstract: 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 $\Vert x_{n}\Vert \rightarrow \Vert x\Vert$, 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 Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for $C^*$-algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.


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Additional Information

Leslie J. Bunce
Affiliation: Department of Mathematics, University of Reading, Reading RG6 2AX, Great Britain
Email: L.J.Bunce@reading.ac.uk

Antonio M. Peralta
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: aperalta@goliat.ugr.es

DOI: https://doi.org/10.1090/S0002-9939-02-06700-X
Received by editor(s): September 27, 2001
Received by editor(s) in revised form: December 3, 2001
Published electronically: September 5, 2002
Additional Notes: The second author was partially supported by D.G.I.C.Y.T. project no. PB 98-1371, and Junta de Andalucía grant FQM 0199
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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