Functional calculus for certain Banach function algebras

Abstract

In this paper we study the symbolic calculus for a Banach function algebra with certain conditions. First we define a class of Banach function algebras which contains the class of all function algebras and the class of all ultraseparating Banach function algebras. Our purpose is to prove the theorem asserting that if A is a non-trivial Banach function algebra in the class, then only analytic functions can operate on A. It is a generalization of theorems of de Leeuw and Katznelson [6], Bernard [2] and Bernard and Dufresnoy [3].