On the character space of Banach vector-valued function algebras

‎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 $mathscr{A}$ on $X$‎, ‎conditions under which the character space $mathfrak{M}(mathscr{A})$‎ ‎is homeomorphic to $mathfrak{M}(mathfrak{A})times mathfrak{M}(A)$ are presented‎, ‎where $mathfrak{A}=C(X) capmathscr{A}$‎ ‎is the subalgebra of $mathscr{A}$ consisting of scalar-valued functions‎. ‎An illustration of the results is given by some examples.