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

Abstract

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

7 pages