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Journal of Operator Theory

Volume 63, Issue 2, Spring 2010  pp. 417-424.

A characterization of scattered $C^*$-algebras and its applications to $C^*$-crossed products

Authors Masaharu Kusuda
Author institution: Department of Mathematics, Faculty of Engineering Science, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan

Summary:  It is well known that any scattered $C^*$-algebra is of type I and AF. We give conditions for $C^*$-algebras being of type I or AF to be scattered. In particular, it is shown that a $C^*$-algebra $A$ is scattered if and only if $A$ is a type~I $C^*$-algebra satisfying that the center of $A$ is scattered. As an application to a $C^*$-dynamical system $(A, G, \alpha)$ with a compact abelian group $G$, it is shown that the fixed point algebra $A^{\alpha}$ of $A$ under the action $\alpha$ is scattered if and only if so is the $C^*$-crossed product $A\times_\alpha G$.

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