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Reduced ${\bi C}^*$-crossed products by free shifts

Published online by Cambridge University Press:  01 October 1998

MARIE CHODA
Affiliation:
Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara 582, Japan. (e-mail: [email protected])
TOSHIKAZU NATSUME
Affiliation:
Department of Mathematics, College of Science and Engineering, Ritsumeikan University, Kusatsu 525, Japan Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, USA Current address: School of Mathematics, Nagoya Institute of Technology, Showa-ku, Nagoya 466, Japan

Abstract

The free shift $\alpha$ of the reduced free product $C^*$-algebra $A$ is studied from both the analytic and non-commutative ergodic theoretic viewpoints. For an automorphism $\beta$ of $B$, we show that the entropy of $\mathop{\rm Ad}\nolimits u(\alpha \otimes \beta)$ is equal to the entropy of $\mathop{\rm Ad}\nolimits u(\beta)$. We also show that if $B$ is unital, nuclear, and simple, and if the crossed product $B \rtimes_\beta {\Bbb Z}$ is simple and purely infinite, then $(O_\infty \otimes B)\rtimes_{\alpha \otimes \beta} {\Bbb Z}$ is isomorphic to $B \rtimes_\beta {\Bbb Z}$.

Type
Research Article
Copyright
© 1998 Cambridge University Press

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