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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
MathSciNet review: 1070349
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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.

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Article copyright: © Copyright 1992 American Mathematical Society