Stacey has recently characterised the crossed product
A
×
α
N
A{ \times _\alpha }{\mathbf {N}}
of a
C
∗
{C^{\ast }}
-algebra
A
A
by an endomorphism
α
\alpha
as a
C
∗
{C^{\ast }}
-algebra whose representations are given by covariant representations of the system
(
A
,
α
)
(A,\alpha )
. Following work of O’Donovan for automorphisms, we give conditions on a covariant representation
(
π
,
S
)
(\pi ,S)
of
(
A
,
α
)
(A,\alpha )
which ensure that the corresponding representation
π
×
S
\pi \times S
of
A
×
α
N
A{ \times _\alpha }{\mathbf {N}}
is faithful. We then use this result to improve a theorem of Paschke on the simplicity of
A
×
α
N
A{ \times _\alpha }{\mathbf {N}}
.