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

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