Continuity of Banach Algebra Valued Functions

Abstract

Abstract. Let K be a compact Hausdorff space, A a commutative complex Banach algebra with identity and C (A ) the set of characters of A . In this note, we study the class of functions f : K → A such that Ω A • f is continuous, where Ω A denotes the Gelfand representation of A . The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly continuous functions relative to the weak topology σ(A , C (A )), are discussed. We also provide some results on its completeness under the norm defined by |f | = Ω A • f ∞ .