  Given a compact space $X$ and a commutative Banach algebra $A$, the character
spaces of $A$-valued function algebras on $X$ are investigated. The class of
natural $A$-valued function algebras, those whose characters can be described
by means of characters of $A$ and point evaluation homomorphisms, is introduced
and studied. For an admissible Banach $A$-valued function algebra $\mathcal{A}$
on $X$, conditions under which the character space $M(\mathcal{A})$ is
homeomorphic to $M(\mathfrak{A}) \times M(A)$ are presented, where
$\mathfrak{A}=C(X) \cap \mathcal{A}$ is the subalgebra of $\mathcal{A}$
consisting of scalar-valued functions. An illustration of the results is given
by some examples.
