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Transactions of the American Mathematical Society

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A representation theorem and applications to topological groups

Author: J.-M. Belley
Journal: Trans. Amer. Math. Soc. 260 (1980), 267-279
MSC: Primary 43A35; Secondary 28A25, 43A60
MathSciNet review: 570789
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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\vert 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.

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Keywords: Algebras of sets, finitely additive set functions, integral representation of linear functionals, Bohr compactification, almost periodic functions, positive definite functions, Fourier-Stieltjes transforms
Article copyright: © Copyright 1980 American Mathematical Society

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