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Continuity of functions operating on characteristic functions. (English) Zbl 0854.22003

Let \(G\) be a locally compact Abelian group, \(P_0(G)\) the set of all continuous positive definite functions \(\varphi : G \to \mathbb{C}\) with \(\varphi(0) = 1\) (characteristic functions on \(G\)). Denote by \(S_0(G)\) the set of all complex-valued functions \(f\) on \[ D(G) = \{z \in \mathbb{C} : z = \varphi(x), \quad x \in G, \quad \varphi \in P_0(G)\} \] operating on \(P_0(G)\), i.e. such that \(f \circ \varphi \in P_0(G)\) for each \(\varphi \in P_0(G)\). It is known that the functions from \(S_0(G)\) are continuous on the set \(\text{int} D(G)\) for every infinite group \(G\) and that for a discrete group \(G\) the functions from \(S_0(G)\) are discontinuous. In 1977, C. C. Graham asked how the continuity of the functions from \(S_0(G)\) on the (whole) set \(D(G)\) depends on the topology on \(G\). The author proves that all \(f \in S_0(G)\) are continuous on \(D(G)\) if and only if \(G\) is a non-discrete group. Using this result he gives a representation of the functions from \(S_0(G)\) for two special classes of (non-discrete) groups \(G\).

MSC:

22B10 Structure of group algebras of LCA groups
22A10 Analysis on general topological groups
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References:

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