Let
N⊂
M be a pair of factors. Associated with a conditional expectation
E from
M onto
N with finite index, we introduce the canonical shift Γ on the von Neumann algebra
A, with the canonical state φ, generated by the tower of relative commutants for the basic constructions iterated from
E. Related with the minimum index [
M:
N]
0, we investigate the entropy
hφ(Γ) of Γ and the entropy
Hφ(
A|Γ(
A)) of
A relative to the subalgebra Γ(
A). The inequalities
hφ(Γ)≤log[
M:
N]
0 and ½
Hφ(
A|Γ(
A))≤log[
M:
N]
0 hold in general. Furthermore when
E has the minimum index and
N⊂
M has finite depth, we establish
hφ(Γ)=½
Hφ(
A|Γ(
A))=log[
M:
N]
0.
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