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Proceedings of the American Mathematical Society

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Entropy for automorphisms of free groups


Author: Marie Choda
Journal: Proc. Amer. Math. Soc. 134 (2006), 2905-2911
MSC (2000): Primary 46L55; Secondary 46L40, 46L89
DOI: https://doi.org/10.1090/S0002-9939-06-08318-3
Published electronically: April 11, 2006
MathSciNet review: 2231614
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Abstract: Let $\sigma$ be the automorphism of the free group $F_\infty$ which is arising from a permutation of the free generators of $F_\infty .$ The $\sigma$ naturally induces the automorphism $\hat \sigma$ of the reduced $C^*$-algebra $C^*_r(F_\infty ),$ and also the automorphism $\bar {\hat \sigma }$ of the group factor $L(F_\infty ).$ We show that the Brown-Germain entropy $ha(\sigma )$ is zero. This implies that the Brown-Voiculescu topological entropy $ht(\hat \sigma ),$ the Connes-Narnhofer-Thirring dynamical entropy $h_\phi (\hat \sigma )$ and the Connes-Størmer entropy $H(\bar {\hat \sigma } )$ are all zero.


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Additional Information

Marie Choda
Affiliation: Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara 582-8582, Japan
Email: marie@cc.osaka-kyoiku.ac.jp

Keywords: Entropy, free group, $C^*$-algebra, amenability
Received by editor(s): February 16, 2005
Received by editor(s) in revised form: April 20, 2005
Published electronically: April 11, 2006
Additional Notes: The author was supported in part by JSPS Grant #14540205.
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.