Entropy for canonical shifts

Choda, Marie

doi:10.1090/S0002-9947-1992-1070349-0

Entropy for canonical shifts

Author: Marie Choda
Journal: Trans. Amer. Math. Soc. 334 (1992), 827-849
MSC: Primary 46L55; Secondary 46L35
DOI: https://doi.org/10.1090/S0002-9947-1992-1070349-0
MathSciNet review: 1070349
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a $^{\ast }$-endomorphism $\sigma$ of an injective finite von Neumann algebra $A$ , we investigate the relations among the entropy $H(\sigma )$ for $\sigma$ , the relative entropy $H(A|\sigma (A))$ of $\sigma (A)$ for $A$ , the generalized index $\lambda (A,\sigma (A))$, and the index for subfactors. As an application, we have the following relations for the canonical shift $\Gamma$ for the inclusion $N \subset M$ of type $\text {II}_{1}$ factors with the finite index $[M:N]$, \[ H(A|\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],\] where $A$ is the von Neumann algebra generated by the two of the relative commutants of $M$. In the case of that $N \subset M$ has finite depth, then all of them coincide.

Marie Choda, Shifts on the hyperfinite ${\rm II}_1$-factor, J. Operator Theory 17 (1987), no. 2, 223–235. MR 887220

---, Entropy for $^{\ast }$ -endomorphisms and relative entropy for subalgebras, J. Operator Theory (to appear).

A. Connes and E. Stornier, Entropy for automorphism of ${\text {II}}_1$ von Neumann algebras, Acta Math. 134 (1975), 288-306.

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
Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI https://doi.org/10.1007/BF01389127
Roberto Longo, Simple injective subfactors, Adv. in Math. 63 (1987), no. 2, 152–171. MR 872351, DOI https://doi.org/10.1016/0001-8708%2887%2990051-X
R. Longo, Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split $W^{\ast } $-inclusions, Invent. Math. 76 (1984), no. 1, 145–155. MR 739630, DOI https://doi.org/10.1007/BF01388497
Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
Mihai Pimsner and Sorin Popa, Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988), no. 1, 127–133. MR 965748, DOI https://doi.org/10.1090/S0002-9947-1988-0965748-8
Sorin Popa, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), no. 1, 27–48. MR 720738, DOI https://doi.org/10.1016/0001-8708%2883%2990033-6

---, Classification of subfactors: The reduction to commuting squares, Preprint.

Robert T. Powers, An index theory for semigroups of $^*$-endomorphisms of ${\scr B}({\scr H})$ and type ${\rm II}_1$ factors, Canad. J. Math. 40 (1988), no. 1, 86–114. MR 928215, DOI https://doi.org/10.4153/CJM-1988-004-3
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
Hisaharu Umegaki, Conditional expectation in an operator algebra, Tohoku Math. J. (2) 6 (1954), 177–181. MR 68751, DOI https://doi.org/10.2748/tmj/1178245177

H. Wenzl, Representation of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.