A holomorphic functional calculus for vector-valued function algebras

Abstract

In this paper, an infinite-dimensional holomorphic functional calculus is established for vector-valued function algebras. Let E be a commutative unital Banach algebra, A be an admissible E-valued function algebra, and . Denoted by , the E-valued spectrum of f is defined as a compact subset of E. It is shown that if a function is holomorphic on a neighbourhood then . As a result, it is proved that the only admissible E-valued uniform algebras are of the form where A runs over the complex uniform algebras.