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