A representation theorem and applications to topological groups

Abstract:

We show that, given a set S dense in a compact Hausdorff space X and a complex-valued bounded linear functional $\Lambda$ on the space $C(X)$ of continuous complex-valued functions on X with uniform norm, there exist an algebra ${\mathcal {A}}$ of sets in S and a unique bounded finitely additive set function $\mu : {\mathcal {A}} \to {\textbf {C}}$ which is inner and outer regular with respect to the zero and cozero sets respectively and such that $\int _s {f\left | S \right . d\mu }$ exists and is equal to $\Lambda (f)$ for all $f \in C(X)$. In the context of topological groups, this theorem permits us to obtain (1) a concrete representation theorem for bounded complex-valued linear functionals on the space of almost periodic functions with uniform norm, (2) a representation theorem for (not necessarily continuous) positive definite functions, (3) a characterization of the space B of finite linear combinations of positive definite functions, and (4) a necessary and sufficient condition to have a linear transformation from B to B.