The alternative Dunford-Pettis property in $C^*$-algebras and von Neumann preduals
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- by Leslie J. Bunce and Antonio M. Peralta PDF
- Proc. Amer. Math. Soc. 131 (2003), 1251-1255 Request permission
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 $\|x_{n}\| \rightarrow \|x\|$, 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.References
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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
- MR Author ID: 666723
- ORCID: 0000-0003-2528-8357
- Email: aperalta@goliat.ugr.es
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1948117