Journal of Operator Theory

Volume 49, Issue 1, Winter 2003 pp. 173-183.

$C^*$-crossed products of $C^*$-algebras with the weak Banach-Saks property

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

Summary: Let $(A, G, \alpha)$ be a $C^*$-dynamical system. In Section 2, we first treat the discrete group action case. We suppose that $G$ acts freely on the spectrum of $A$. Then it is shown that $A$ has the weak Banach-Saks property, if and only if $G$ is discrete and the $C^*$-crossed product $A\times_\alpha G$ has the weak Banach-Saks property. In Section 3, we shall consider the compact group action case. Let $G$ be a compact group and consider the following conditions (1)--(3): (1) $A$ has the weak Banach-Saks property; (2) $A\times_\alpha G$ has the weak Banach-Saks property; (3) the fixed point algebra $A^{\alpha}$ of $A$ has the weak Banach-Saks property. Then it is shown that we have (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3). Furthermore we suppose that $G$ is (compact) abelian. Then it is shown that the implication (3) $\Longrightarrow$ (2) holds, and that if $A$ is of type I and if $\alpha$ is pointwise unitary, the implication (2) $\Longrightarrow$ (1) holds.

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