On 1-cocycles induced by a positive definite function on a locally compact abelian group

For $φ$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $π_φ$ associated to $φ$ by the GNS construction, on the other hand the probability measure $μ_φ$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,π_φ)$ and reduced 1-cohomology $\bar{H}^1(G,π_φ)$. For example, $\bar{H}^1(G,π_φ)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $μ_φ(1_G)=0$, where $1_G$ is the trivial character of $G$.

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