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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