For each of the radical Banach algebras
L
1
(
0
,
1
)
{L^1}(0,1)
and
L
1
(
w
)
{L^1}(w)
an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of
L
1
(
0
,
1
)
{L^1}(0,1)
and
L
1
(
w
)
{L^1}(w)
endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if
|
|
|
⋅
|
|
|
p
||| \cdot ||{|_p}
denotes the norm of
B
(
L
p
(
0
,
1
)
B({L^p}(0,1)
,
L
1
(
0
,
1
)
)
{L^1}(0,1))
,
1
>
p
≤
∞
1 > p \leq \infty
, then the group of automorphisms of
L
1
(
0
,
1
)
{L^1}(0,1)
topologized by
|
|
|
⋅
|
|
|
p
||| \cdot ||{|_p}
is arcwise connected. It is shown that every automorphism
θ
\theta
of
L
1
(
0
,
1
)
{L^1}(0,1)
is of the form
θ
=
e
λ
d
lim
e
q
n
(
BSO
)
\theta = {e^{\lambda d}}{\operatorname {lim}}{e^{qn}}({\text {BSO}})
, where each
q
n
{q_n}
is a quasinilpotent derivation. It is shown that the group of principal automorphisms of
l
1
(
w
)
{l^1}(w)
under operator norm topology is arcwise connected, and every automorphism has the form
e
i
α
d
(
e
λ
d
e
D
e
−
λ
d
)
−
{e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }
, where
α
∈
R
\alpha \in {\mathbf {R}}
,
λ
>
0
\lambda > 0
, and
D
D
is a derivation, and where
(
e
λ
d
e
D
e
−
λ
d
)
−
{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }
denotes the extension by continuity of
e
λ
d
e
D
e
−
λ
d
{e^{\lambda d}}{e^D}{e^{ - \lambda d}}
from a dense subalgebra of
l
1
(
w
)
{l^1}(w)
to
l
1
(
w
)
{l^1}(w)
.