Range Transformations on a Banach Function Algebra

We study the range transformations 0p(/fD, Reß) and Op(/fD, B) for Banach function algebras A and B. As a special instance, the harmonicity of functions in Op(/fD, Re A) for a nontrivial function algebra A is established and is compared with previous investigations of Op( An, A) and Op((Re/l);, (Re/1)) for an interval /. In §2 we present some results on Op(AD, B) and use them to show that functions in Op'(AD, B) are analytic for certain Banach function algebras. Introduction. We say that A is a Banach function algebra on a compact Hausdorff space X if A is a unital subalgebra of C(X) which is a Banach algebra with the norm A^(-) such that A separates the points of A. A function algebra on A is a Banach function algebra on A with the uniform norm as the Banach algebra norm. We denote by || • ||v the uniform norm on a subset Y of X. If £ is a normed space, we denote by £ the set of all bounded sequences in £. If A is a Banach function algebra, then Ä is a unital Banach algebra lying in C( A), where A is the Stone-Cech compactification of the direct product A X A of the discrete space A of all positive integers and A, with the Banach algebra norm ÑA(f) = sup{NA(fn): n = 1,2,...} for /= (/„) in A. In this paper we denote Banach function algebras on a compact Hausdorff space A by A or B unless we mention it. Definition 1. We say that A is ultraseparating if Ä separates the points of X. Definition 2. Let x be a point in X. We denote: Ax= {f£A:f(x) = 0}, Q(AX)= {xeX:f(x) = OVf<=Ax},