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For a normalized positive definite function on a locally compact abelian group , let be the unitary representation associated to by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology and reduced 1-cohomology . For example, if and only if either or , where is the trivial character of and is the probability measure on the Pontryagin dual associated to by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem...
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