On algebras of Banach algebra-valued bounded continuous functions

Abstract

Let $X$ be a completely regular Hausdorff space. We denote by $C(X,A)$ the algebra of all continuous functions on $X$ with values in a complex commutative unital Banach algebra $A$. Let $C_{b}(X,A)$ be its subalgebra consisting of all bounded continuous functions and endowed with the uniform norm. In this paper, we give conditions equivalent to the density of a natural continuous image of $X\times \mathcal {M}(A)$ in the maximal ideal space of $C_{b}(X,A)$.