## Entropy for automorphisms of free groups

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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.## References

- C. Anantharaman-Delaroche and J. Renault,
*Amenable groupoids*, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR**1799683** - Nathanial P. Brown,
*Topological entropy in exact $C^\ast$-algebras*, Math. Ann.**314**(1999), no. 2, 347–367. MR**1697449**, DOI 10.1007/s002080050298 - Nathanial P. Brown and Marie Choda,
*Approximation entropies in crossed products with an application to free shifts*, Pacific J. Math.**198**(2001), no. 2, 331–346. MR**1835512**, DOI 10.2140/pjm.2001.198.331 - Nathanial P. Brown and Emmanuel Germain,
*Dual entropy in discrete groups with amenable actions*, Ergodic Theory Dynam. Systems**22**(2002), no. 3, 711–728. MR**1908551**, DOI 10.1017/S0143385702000366 - Marie Choda,
*Dynamical entropy for automorphisms of exact $C^*$-algebras*, J. Funct. Anal.**198**(2003), no. 2, 481–498. MR**1964548**, DOI 10.1016/S0022-1236(02)00033-2 - A. Connes and E. Størmer,
*Entropy for automorphisms of $II_{1}$ von Neumann algebras*, Acta Math.**134**(1975), no. 3-4, 289–306. MR**454657**, DOI 10.1007/BF02392105 - A. Connes, H. Narnhofer, and W. Thirring,
*Dynamical entropy of $C^\ast$ algebras and von Neumann algebras*, Comm. Math. Phys.**112**(1987), no. 4, 691–719. MR**910587**, DOI 10.1007/BF01225381 - Kenneth J. Dykema,
*Topological entropy of some automorphisms of reduced amalgamated free product $C^*$-algebras*, Ergodic Theory Dynam. Systems**21**(2001), no. 6, 1683–1693. MR**1869065**, DOI 10.1017/S0143385701001808 - David Kerr and Claudia Pinzari,
*Noncommutative pressure and the variational principle in Cuntz-Krieger-type $C^*$-algebras*, J. Funct. Anal.**188**(2002), no. 1, 156–215. MR**1878635**, DOI 10.1006/jfan.2001.3835 - Eberhard Kirchberg and Simon Wassermann,
*Exact groups and continuous bundles of $C^*$-algebras*, Math. Ann.**315**(1999), no. 2, 169–203. MR**1721796**, DOI 10.1007/s002080050364 - Narutaka Ozawa,
*Amenable actions and exactness for discrete groups*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 8, 691–695 (English, with English and French summaries). MR**1763912**, DOI 10.1016/S0764-4442(00)00248-2 - Erling Størmer,
*Entropy of some automorphisms of the $\textrm {II}_1$-factor of the free group in infinite number of generators*, Invent. Math.**110**(1992), no. 1, 63–73. MR**1181816**, DOI 10.1007/BF01231324 - Erling Størmer,
*States and shifts on infinite free products of $C^*$-algebras*, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 281–291. MR**1426846**, DOI 10.1103/physrevb.55.r7351 - E. Størmer,
*A survey of noncommutative dynamical entropy*, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 147–198. MR**1878883**, DOI 10.1007/978-3-662-04825-2_{2} - Dan Voiculescu,
*Dynamical approximation entropies and topological entropy in operator algebras*, Comm. Math. Phys.**170**(1995), no. 2, 249–281. MR**1334396**, DOI 10.1007/BF02108329

## Additional Information

**Marie Choda**- Affiliation: Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara 582-8582, Japan
- Email: marie@cc.osaka-kyoiku.ac.jp
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2231614