Entropy for canonical shifts

Choda, Marie

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

Entropy for canonical shifts
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by Marie Choda

Trans. Amer. Math. Soc. 334 (1992), 827-849

DOI: https://doi.org/10.1090/S0002-9947-1992-1070349-0

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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 $H(A|\sigma (A))$ 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.

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