Let
σ
\sigma
be the automorphism of the free group
F
∞
F_\infty
which is arising from a permutation of the free generators of
F
∞
.
F_\infty .
The
σ
\sigma
naturally induces the automorphism
σ
^
\hat \sigma
of the reduced
C
∗
C^*
-algebra
C
r
∗
(
F
∞
)
,
C^*_r(F_\infty ),
and also the automorphism
σ
^
¯
\bar {\hat \sigma }
of the group factor
L
(
F
∞
)
.
L(F_\infty ).
We show that the Brown-Germain entropy
h
a
(
σ
)
ha(\sigma )
is zero. This implies that the Brown-Voiculescu topological entropy
h
t
(
σ
^
)
,
ht(\hat \sigma ),
the Connes-Narnhofer-Thirring dynamical entropy
h
ϕ
(
σ
^
)
h_\phi (\hat \sigma )
and the Connes-Størmer entropy
H
(
σ
^
¯
)
H(\bar {\hat \sigma } )
are all zero.