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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