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Translation-invariant function algebras on compact abelian groups. (English) Zbl 0892.46062
The author characterizes certain algebras of the kind mentioned in the title in terms of orderings on the dual group: Let \(G\) be a compact abelian group with normalized Haar measure \(m\) and dual group \(\Gamma\). For a translation-invariant order \(\leq\) on \(\Gamma\) let \(D(\Gamma,\leq)\) denote the closed linear span of the positive cone \(\{\chi \mid 1\leq \chi\}\). The author shows that for a translation-invariant function algebra \(A\) on \(G\) the following conditions are equivalent: 1) \(A= D(\Gamma,\leq)\) for some \(\leq\). 2) \(A\) is an antisymmetric Dirichlet algebra. 3) \(A\) is logmodular and \(m\) is multiplicative. 4) \(m\) is the unique representing measure for a character on \(A\). 5) \(m\) is an extremal representing measure for a character of \(A\).
MSC:
46J10 Banach algebras of continuous functions, function algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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