Homomorphisms on some function algebras

Abstract

For an algebra A of continuous real-valued functions on a topological space X, the question of whether every algebra homomorphism is a point evaluation for a point in X is considered. A variety of results are provided, such as the following. Let X be completely regular and A⊂C(X) a subalgebra with unit which is closed under bounded inversion and separates points and closed sets. Then every homomorphism is a point evaluation for a point in X if and only if, for each point z in the Stone-Čech compactification of X and not in X, there exists a function in A whose extension to z is infinite. Examples are considered and further results for the case of functions on a Banach space are discussed