On the character space of Banach vectorvalued 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 A on X, conditions under which the character space M(A) is homeomorphic to M(A) × M(A) are presented, where A = C(X) ∩ A is the subalgebra of A consisting of scalar-valued functions. An illustration of the results is given by some examples.