Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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\vert\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 II$ _{1}$ factors with the finite index $ [M:N]$,

$\displaystyle H(A\vert\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],$

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.

References [Enhancements On Off] (What's this?)

  • [1] M. Choda, Shifts on the hyperfinite $ {\text{II}}_1$-factor, J. Operator Theory 17 (1987), 223-235. MR 887220 (88g:46075)
  • [2] -, Entropy for $ ^{\ast} $ -endomorphisms and relative entropy for subalgebras, J. Operator Theory (to appear).
  • [3] A. Connes and E. Stornier, Entropy for automorphism of $ {\text{II}}_1$ von Neumann algebras, Acta Math. 134 (1975), 288-306.
  • [4] A Connes, H. Narnhofer, and W. Thirring, Dynamical entropy of $ {C^{\ast} }$-algebras and von Neumann algebras, Comm. Math. Phys. 112 (1987), 691-719. MR 910587 (89b:46078)
  • [5] F. Goodman, P. de la Harpe, and V. Jones, Coxeter graphs and towers of algebras, Math. Sci. Res. Inst. Publ. 14, Springer-Verlag, 1989. MR 999799 (91c:46082)
  • [6] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. MR 696688 (84d:46097)
  • [7] R. Longo, Simple injective subfactors, Adv. in Math. 63 (1987), 152-171. MR 872351 (88i:46082)
  • [8] -, Solution of the factorial Stone- Weierstrass conjecture. An application of the theory of standard split $ {W^{\ast} }$-inclusions, Invent. Math. 76 (1984), 145-155. MR 739630 (85m:46057a)
  • [9] A. Ocneanu, Quantized groups, string algebras, and Galois theory for algebras, Preprint. MR 996454 (91k:46068)
  • [10] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106. MR 860811 (87m:46120)
  • [11] -, Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988), 127-134. MR 965748 (89k:46073)
  • [12] S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), 27-48. MR 720738 (85h:46084)
  • [13] -, Classification of subfactors: The reduction to commuting squares, Preprint.
  • [14] R. T. Powers, An index theory for semigroups of $ ^{\ast} $ -endomorphisms of $ B(H)$ and type $ {\text{II}}_1$-factors, Canad. J. Math. 40 (1988), 86-114. MR 928215 (89f:46116)
  • [15] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, 1979. MR 548728 (81e:46038)
  • [16] H. Umegaki, Conditional expectations in an operator algebra, Tôhoku Math. J. 6 (1954), 358-362. MR 0068751 (16:936b)
  • [17] H. Wenzl, Representation of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L55, 46L35

Retrieve articles in all journals with MSC: 46L55, 46L35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1070349-0
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society