Journal of Operator Theory

Volume 60, Issue 2, Fall 2008 pp. 301-316.

On Mazur's property and property (X)

Authors: Matthias Neufang
Author institution: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S5B6, Canada

Summary: We give a complete characterization of those von Neumann algebras whose preduals have Mazur's Property. We further show that for preduals of von Neumann algebras, Mazur's Property is actually equivalent to Property~(X) which was first studied by Godefroy and Talagrand in \cite{god-tal}. Moreover, we introduce and study natural generalizations of the latter properties to the level of arbitrary cardinal numbers $\kappa$, as suggested in \cite{poly} for Property~(X). In particular, using Edgar's partial ordering of Banach spaces \cite{edg}, we prove that Property~(X) of level $\kappa$ only differs from the original one in the case where $\kappa$ is a measurable cardinal number. Several applications of our results to some concrete spaces such as $L_1(\G)$ for a locally compact group$\G$ and the space of trace class operators $\mathcal{T}(\H)$ on a Hilbert space are also discussed.

Keywords: Mazur's Property, Property~(X), predual of von Neumann algebra, measurable cardinal.

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