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 [11]. Moreover, we introduce and study natural generalizations of the latter properties to the level of arbitrary cardinal numbers Κ, as suggested in [9] for Property (X). In particular, using Edgar's partial ordering of Banach spaces [4], we prove that Property (X) of level Κ only differs from the original one in the case where Κ is a measurable cardinal number. Several applications of our results to some concrete spaces such as L1 (G) for a locally compact group G and the space of trace class operators T(H) on a Hubert space are also discussed.

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Journal of Operator Theory
School of Mathematics and Statistics

Neufang, M. (2008). On mazur's property and property (X). Journal of Operator Theory, 60(2), 301–316.