Averages of Continuous Functions on Countable Spaces

Let X = {x1, x2, ...} be a countably infinite topological space; then the space C*(X) of all bounded real-valued continuous functions f may be regarded as a space of sequences (f(x1), f(x2), ...). It is well known [7, p. 54] that no regular (Toeplitz) matrix can sum all bounded sequences. On the other hand, if (x1, x2, ...) converges in X (to xm), then every regular matrix sums all f in C*(X) (to f(xm)). The main result of this paper is that if a regular matrix sums all f in C*(X) then it sums f to Σαnf(xn), for some absolutely convergent series Σαn. We use this to show that no regular matrix can sum all of C*(X) if X is extremally disconnected (the closure of every open set is open). This extends a theorem of W. Rudin [6], which has an equivalent hypothesis (X is embeddable in the Stone-Cech compactification βN of a discrete space) and concludes that not all f in C*(X) are Cesàro summable. For any continuous linear functional φ on C*(X) one has a ( Riesz ) representation φ(f) = ∫fdμ, where μ is a Radon measure on βX. Our main result is just that X supports μ; μ is forced to be atomic since X is countable. We show further that X has a subset T, the set of heavy points, such that the functionals we are concerned with correspond exactly to measures μ supported by T with μ(T) = 1. Our knowledge of T is limited; it will be summarized elsewhere