Separable $C^{\ast}$-Algebras and weak$^{\ast}$-fixed point property

It is shown that the dual $\hat{A}$ of a separable $C^{\ast}$-algebra $A$ is discrete if and only if its Banach space dual has the weak$^{\ast}$-fixed point property. We prove further that these properties are equivalent to the uniform weak$^{\ast}$ Kadec-Klee property of $A^{\ast}$ and to the coinc...

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Bibliographic Details
Main Authors Fendler, Gero, Leinert, Michael
Format Journal Article
LanguageEnglish
Published 22.03.2013
Subjects
Mathematics - Operator Algebras
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Summary:It is shown that the dual $\hat{A}$ of a separable $C^{\ast}$-algebra $A$ is discrete if and only if its Banach space dual has the weak$^{\ast}$-fixed point property. We prove further that these properties are equivalent to the uniform weak$^{\ast}$ Kadec-Klee property of $A^{\ast}$ and to the coincidence of the weak$^{\ast}$ topology with the norm topology on the pure states of $A$.
DOI:10.48550/arxiv.1303.5557