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Entropy for extensions of Bernoulli shifts

Published online by Cambridge University Press:  14 October 2010

Marie Choda
Marie Choda
Affiliation:
Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara 582, Japan
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Abstract

We give a condition for automorphisms α and β on finite von Neumann algebras which induces the tensor product formula for entropies: H(α ⊗ β) = H(α) + H(β). As an application, the Bernoulli shift (1/n, 1/n, …, 1/n) has extensions to ergodic outer automorphisms {αk; k = 1,2, …} on the hyperfinite II1 factor R with the entropies H(αk) = (1/2)kn log n.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1197 - 1206
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1] Besson, O.. On the entropy in II1 von Neumann algebras. Ergod. Th. & Dynam. Sys. 1 (1981), 419–429.CrossRefGoogle Scholar
[2] Bures, D. and Yin, H.-S.. Shifts on the hyperfinite factor of II1. J. Operator Theory 20 (1988), 91–106.Google Scholar
[3] Choda, M.. Shifts on the hyperfinite II1 factor. J. Operator Theory 17 (1987), 223–235.Google Scholar
[4] Choda, M.. Entropy for *-endomorphisms and relative entropy for subalgebras. J. Operator Theory 25 (1991), 125–140.Google Scholar
[5] Choda, M.. Entropy for canonical shifts. Trans. Amer. Math. Soc. 334 (1992), 827–849.CrossRefGoogle Scholar
[6] Choda, M. and Hiai, F.. Entropy for canonical shifts II. Publ. RIMS. Kyoto Univ. 27 (1991), 461–489.Google Scholar
[7] Choda, M.. Conjugate but not inner conjugate subfactors. Proc. Amer. Math. Soc. 124 (1996), 147–153.CrossRefGoogle Scholar
[8] Connes, A. and Størmer, E.. Entropy for automorphisms of von Neumann algebras. Ada Math. 134 (1975), 188–306.Google Scholar
[9] Connes, A. and Størmer, E.. A connection between the classical and the quantum mechanical entropy. Operator Algebras and Group Representations (Monographs and Studies in Mathematics 17). Pitman, 1984, pp. 113–123.Google Scholar
[10] Connes, A., Narnhofer, H. and Thirring, W.. Dynamical entropy of von Neumann algebras. Commun. Math. Phys. 112 (1987), 691–719.Google Scholar
[11] Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.CrossRefGoogle Scholar
[12] Enomoto, M., Choda, M. and Watatani, Y.. Generalized Powers' binary shifts on the hyperfinite II1 factor. Math. Japonica 33 (1988), 831–843.Google Scholar
[13] Jones, V. F. R.. Index for subfactors. Invent. Math. 72 (1983), 1–25.CrossRefGoogle Scholar
[14] Narnhofer, H., Størmer, E. and Thirring, W.. C*-dynamical systems for which the tensor product formula for entropy fails. Ergod. Th. & Dynam. Sys. 15 (1995), 961–968.Google Scholar
[15] Pimsner, M. and Popa, S.. Entropy and index for subfactors. Ann. Sci. Ecole Norm. Sup. 19 (1986), 57–106.Google Scholar
[16] Popa, S.. Notes on Cartan subalgebras in type II1 factors. Math. Scand. 57 (1985), 171–188.Google Scholar
[17] Powers, R. T.. An index theory for semigroups of *-endomorphisms of B(H) and type II1 factors. Canad. J. Math. 40 (1988), 86–114.CrossRefGoogle Scholar
[18] Price, G. L.. Shifts of integer index on the hyperfinite II1 factor. Pacific J. Math. 132 (1988), 379–390.CrossRefGoogle Scholar
[19] Størmer, E., Voiculescu, D.. Entropy of Bogoliubov automorphisms of the canonical anticommutation relations. Commun. Math. Phys. 133 (1990), 521–542.Google Scholar
[20] Størmer, E.. Entropy of some inner automorphisms of the hyperfinite II1 factor. Inter. J. Math. 4 (1993), 319–322.Google Scholar
[21] Voiclescu, D.. Dynamical approximation entropies and topological entropy in operator algebras. Commun. Math. Phys. 170 (1995), 249–281.CrossRefGoogle Scholar