Let
X
,
Y
X,Y
be locally compact Hausdorff spaces and
M
{\mathcal M}
,
N
{\mathcal N}
be Banach algebras. Let
θ
:
C
0
(
X
,
M
)
→
C
0
(
Y
,
N
)
\theta : C_0(X,{\mathcal M}) \to C_0(Y, {\mathcal N})
be a zero product preserving bounded linear map with dense range. We show that
θ
\theta
is given by a continuous field of algebra homomorphisms from
M
{\mathcal M}
into
N
{\mathcal N}
if
N
{\mathcal N}
is irreducible. As corollaries, such a surjective
θ
\theta
arises from an algebra homomorphism, provided that
M
{\mathcal M}
is a
W
∗
W^*
-algebra and
N
{\mathcal N}
is a semi-simple Banach algebra, or both
M
{\mathcal M}
and
N
{\mathcal N}
are
C
∗
C^*
-algebras.