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 Classes d'espaces de Banach à prédual unique, C. R. Acad. Sci. Paris 292(1981), 323-325. Moreover, we introduce and study natural generalizations of the latter properties to the level of arbitrary cardinal numbers $\kappa$, as suggested by Poly in Nouvelles classes d'espaces de Banach à prédual unique, $\textit{Séminaire d'Analyse Fonctionnelle 1980-1081}$ for Property $($X$)$. In particular, using Edgar's partial ordering of Banach spaces in An ordering for the Banach spaces, $\textit{Pacific J. Math. } \textbf{108}(1983), 83-98$, 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 $L_1(\mathcal{G})$ for a locally compact group $\mathcal{G}$ and the space of trace class operators $\mathcal{T}(\mathcal{H})$ on a Hilbert space are also discussed.
Keywords: Mazur's Property, Property $($X$)$, predual of von Neumann algebra, measurable cardinal.
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