Homomorphisms on some function algebras

Resumen

Suppose that A is an algebra of continuous real functions defined on a topological space X. We shall be concerned here with the problem as to whether every nonzero algebra homomorphism f: A ? R is given by evaluation at some point of X, in the sense that there exists some a in X such that f(f) = f(a) for every f in A. The problem goes back to the work of Michael [19], motivated by the question of automatic continuity of homomorphisms in a symmetric *-algebra. More recently, the problem has been considered by several authors, mainly in the case of algebras of smooth functions: algebras of differentiable functions on a Banach space in [2], [11], [13] and [14]; algebras of differentiable functions on a locally convex space in [3], [4], [5] and [6], and algebras of smooth functions in the abstract context of smooth spaces in [18]. We shall be interested both in the general case and in the case of functions on a Banach space. This report is based on the results obtained in [8].