Function algebras on which homomorphisms are point evaluations on sequences

Abstract

In the study of the spectrum of a subalgebra A of C(X), where X is a completely regular Hausdorff space, a key question is, whether each homomorphism φ{symbol}:A→R has the point evaluation property for sequences in A, that is whether, for each sequence (fn ) in A, there exists a point a in X such that φ{symbol}(fn )=fn (a) for all n. In this paper it is proved that all algebras, which are closed under composition with functions in C∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the space Cm (E) (m=0,1,...,∞) of Cm -functions on any real locally convex space E. This result yields in a trivial manner that each homomorphism φ{symbol} on A is a point evaluation, if X is Lindelöf or if A contains a sequence which separates points in X. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem. © 1991 Springer-Verlag.