The main result in this paper states that the second order linear
B
∗
{B^*}
-algebra differential equation
(
p
(
t
)
y
′
)
+
q
(
t
)
y
=
0
(p(t)y’) + q(t)y = 0
, where
p
(
t
)
p(t)
is positive and
q
(
t
)
q(t)
is Hermitian for each
t
t
, is nonoscillatory on
[
t
0
,
∞
)
[{t_0},\infty )
if the scalar equation
(
‖
p
−
1
(
t
)
‖
−
1
W
′
)
′
+
‖
q
(
t
)
‖
W
=
0
({\left \| {{p^{ - 1}}(t)} \right \|^{ - 1}}W’)’ + \left \| {q(t)} \right \|W = 0
is nonoscillatory on
[
t
0
,
∞
)
[{t_0},\infty )
. Consequently, every criterion on nonoscillation in the scalar case automatically produces another one in the
B
∗
{B^*}
-algebra case.